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Full text of "Transistors"

lttttt?ratig 
of JFlarffca 
ffithrarUa 



Engineering 

& Physics 

Library 




©Iji? (gift of 
Eugene Chenette 



Digitized by the Internet Archive 
in 2013 



http://archive.org/details/transistorsOOIecr 



Transistors 



by 

Dennis Le Croissette 

Jet Propulsion Laboratory 
California Institute of Technology 



Prentice-Hall, Inc. 

Englewood Cliffs, N. J. 
1963 



PRENTICE-HALL INTERNATIONAL, INC., London 

PRENTICE-HALL OF AUSTRALIA, PTY., LTD., Sydney 

PRENTICE-HALL OF FRANCE, S.A.R.L., Paris 

PRENTICE-HALL OF JAPAN, INC., Tokyo 

PRENTICE-HALL DE MEXICO, S.A., MeXlCO City 

PRENTICE-HALL OF CANADA, LTD., Toronto 





Prentice-Hall Electrical Engineering Series 
William L. Everitt, Editor 



© 1963 by Prentice-Hall, Inc., Englewood Cliffs, N.J. 
All rights reserved. No part of this book may be re- 
produced in any form, by mimeograph or any other 
means, without permission in writing from the publisher. 

Library of Congress Catalog Card Number: 63-7561 



Printed in the United States of America 



93017-C 



Preface The aim of this book is to present and explain the 
basic principles of semiconductor electronics as 
they apply to junction transistors. The introduc- 
tory approach makes it suitable for use as an un- 
dergraduate text in electrical engineering. It will 
also be helpful to engineers in industry who require 
a knowledge of transistors. 

The fundamentals of semiconductor devices 
are introduced here from a physical viewpoint. In 
my opinion, this is the best preparation for later 
work in electronic circuitry where device limita- 
tions must be considered. Moreover, vacuum tube 



electronics is relatively easy to understand after 
these concepts have been developed. This opinion 
is supported by experience at Drexel Institute of 
Technology where part of this material was used 
as the first course in electronics. 

Part 1 of this book is concerned with the flow 
of electrons and holes in semiconducting materials, 
leading to an understanding of current flow in p-n 
junctions and transistors. 

In Part 2, the "black box" concept of the tran- 
sistor is developed. Analysis is restricted to low 
frequency operation of the device, and hybrid (or 
h) parameter representation is used. Direct-current 
operation of the transistor is also considered. 

Part 3 consists mainly of an analysis of the 
high frequency and pulse operation of the tran- 
sistor. The hybrid-jt circuit is derived and used to 
determine the high frequency response of the com- 
mon emitter stage. Pulse operation is described 
from the charge storage point of view. In conclu- 
sion, the tunnel diode is discussed. 

It is apparent that this book, designed pri- 
marily for engineers, cannot derive all the physical 
relationships necessary for an analysis of the tran- 
sistor. The references at the end of each chapter 
assist the reader to find the derivations he requires. 

I feel strongly that this textbook is not suitable 
for a course oriented towards solving numerical 
problems. Therefore, many descriptive questions 
have been given to encourage students to review 
and rephrase the material. 

I would like to acknowledge the assistance of 
John Yarwood of London, England and Eugene J. 
Rosenbaum of Philadelphia, Pa. who read part of 
the manuscript while it was in preparation. My 
former colleagues at Drexel Institute of Technol- 
ogy have made a number of helpful comments. I 
am also grateful to Albert F. Fuchs for contribut- 
ing many of the problems and to Mrs. Noreen J. 
Reyes for typing most of the manuscript. 

DENNIS LE CROISSETTE 



Contents PART ONE: the motion of electrons and holes 

IN SEMICONDUCTING MATERIALS 
1 Electrical Conduction in Semiconductors 3 

1-1 Introduction, 4. 1-2 Simple theory of the hydrogen 
atom, 4. 1-3 Bohr's theory of the hydrogen atom, 5. 
1-4 Energy of the electron orbits of the hydrogen atom, 7. 
1-5 Transitions between the orbits, 9. 1-6 Energy level 
representation, 10. 1-7 Limitations of the Bohr theory, 
11. 1-8 The wave nature of matter, 11. 1-9 Wave- 
mechanical theory of atomic structure, 14. 1-10 The 
atomic table, 15. 1-11 Chemical valence and atomic 
binding, 17. 1-12 Crystals, 18. 1-13 Electron energy 
levels in crystals: band theory, 20. 1-14 Insulators, 24. 
1-15 Conductors, 25. 1-16 Semiconductors, 25. 1-17 
Electrons and positive holes, 26. 1-18 Mobility, 28. 1-19 
p- and n-type semiconductors, 29. 1-20 Generation and 
recombination, 35. 



A Electrons and Holes in Semiconductors 43 

2-1 The density of states in an energy band, 44. 2-2 En- 
ergy state densities in semiconductors and the effect of 
temperature, 47. 2-3 Intrinsic semiconductor, 51. 2-4 
rc-type semiconductor, 54. 2-5 p-type semiconductor, 55. 
2-6 Variation of the Fermi level with temperature, 56. 
2-7 The law of mass action, 57. 2-8 Mobile and im- 
mobile charges, 58. 2-9 The Hall effect, 61. 

J Junctions between Materials 67 

3-1 Work function in a metal, 68. 3-2 Junctions between 
metals, 70. 3-3 Semiconductor junctions with no applied 
voltage, 72. 3-4 Semiconductor junction with applied 
voltage: junction capacitance, 79. 3-5 Flow of current 
across a p-n junction: the rectifier equation, 82. 3-6 The 
breakdown region of a p-n junction, 86. 

4 The Continuity Equation 89 

4-1 Derivation of the continuity equation, 90. 4-2 Ap- 
plication of the continuity equations to the abrupt p-n 
junction operating with a constant current, 94. 4-3 Cal- 
culation of the charge densities at the edge of the deple- 
tion regions, 99. 4-4 Solution of the continuity equations 
for a reverse-biased abrupt p-n junction, 101. 4-5 Solu- 
tion of the continuity equations for a forward-biased 
abrupt p-n junction, 104. 4-6 A brief discussion of the 
p-n junction, 107. 4-7 The incremental resistance of a 
forward-biased p-n junction, 108. 4-8 Einstein equation, 
108. 

J The Junction Transistor 112 

5-1 The npn transistor, 113. 5-2 The pnp transistor, 116. 
5-3 Characteristics of the current flow across the base 
region, 117. 5-4 The transistor as a device: common 
base circuit, 123. 5-5 The transistor as a device: com- 
mon emitter circuit, 126. 5-6 The manufacture of tran- 
sistors, 128. 



PART TWO: TRANSISTORS AT LOW FREQUENCIES 
AND UNDER D-C CONDITIONS 

The Transistor as a Two Port Network 135 

6-1 General considerations of electronic devices as am- 
plifiers, 136. 6-2 Alternating-current operation of the 
transistor, 139. 6-3 z, y, and h parameters: general com- 
ments, 142. 6-4 Equivalent circuit using z parameters: 
the T circuit, 144. 6-5 Equivalent circuit using y param- 
eters: the ji circuit, 148. 6-6 Equivalent circuit using h 
parameters: a two generator representation, 149. 6-7 
Low frequency measurements on transistors, 150. 



Low Frequency h Parameter Representation 154 

7-1 The small-signal h parameters, 155. 7-2 Analysis of 
the general h parameter equivalent circuit, 156. 7-3 Re- 
lationship of the three sets of h parameters, 159. 7-4 
Comparison of the three configurations, 161. 7-5 Varia- 
tion in h parameters with I E and Vcss, 163. 

Single and Multistage a-c Amplifiers 167 

8-1 The characteristic curves, 168. 8-2 The load line, 
170. 8-3 Operating point stability, 172. 8-4 Practical 
biasing circuits, 178. 8-5 Power amplifiers, 182. 8-6 
Power amplifiers: practical limitations, 187. 8-7 Thermal 
runaway, 189. 8-8 Multistage amplifier, 191. 8-9 Fre- 
quency response of the multistage amplifier, 194. 



PART THREE: FURTHER THEORY OF THE TRANSISTOR, 
AND THE TUNNEL DIODE 

y Physical Characteristics of the Transistor 199 

9-1 Solution of the continuity equation in the base region 
for direct currents, 200. 9-2 Components of alpha, 202. 
9-3 The intrinsic transistor, 203. 9-4 The continuity 
equation in the base region for small alternating signals, 
204. 9-5 Base width modulation: transmission line 
analogy, 208. 9-6 The hybrid-jt representation, 211. 
9-7 Variation of alpha with emitter current, 217. 9-8 
The drift transistor, 220. 9-9 Transistor noise, 223. 
9-10 Surface effects in transistors, 225. 9-11 Punch- 
through, 226. 9-12 Epitaxial films, 226. 

iU High Frequency and Pulse Operation of the Transistor 229 

10-1 The hybrid-jt circuit, 230. 10-2 The Miller effect, 
231. 10-3 Frequency response of the transistor in the 
common emitter connection: hybrid-ji model, 233. 10-4 
Frequency response of the iterative common emitter 
stage: hybrid-jt model, 235. 10-5 Frequency response of 
the transistor in the common base connection: simple 
model, 236. 10-6 Comparison of the hybrid-ji circuit 
with the actual transistor, 238. 10-7 Saturation, 242. 
10-8 Switching the transistor on: use of the hybrid-jt 
model, 244. 10-9 Definition of switching times for the 
transistor: saturation case, 247. 10-10 Charge control of 
the transistor, 249. 10-11 Switching times for a saturated 
transistor, 254. 



11 



The Tunnel Diode 260 

11-1 Quantum mechanical tunneling, 261. 11-2 Energy 
bands of the tunnel diode, 263. 11-3 Forward and re- 
verse characteristics, 265. 11-4 The tunnel diode as a 
circuit element, 268. 



APPENDIX 274 

INDEX 275 



PART 



1 



The 

Motion of 

Electrons and 

Holes in 

Semiconducting 

Materials 



1 



Electrical 

Conduction in 

Semiconductors 



When the vacuum tube was the only practicable 
electronic amplifier, the subject of electronics was 
predominantly concerned with the flow of electrons 
in a vacuum or in a gas at low pressure. In the past 
twenty years, however, considerable attention has 
been focused on the conduction of electricity in 
solid materials. Research in solid-state electronics 
was accelerated by the development of the transistor 
in 1948. Since that date, a study of the processes 
governing current flow in semiconductors has been 
essential for an understanding of electronics. 



4 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

This chapter begins with a review of the basic atomic properties of matter 
leading to the energy band representation of insulators, conductors, and 
semiconductors. Later in this chapter, the fundamental concept of hole and 
electron conduction in a semiconductor is developed, and recombination 
between holes and electrons is briefly discussed. 



1-1 Introduction 

In the past, solid materials were divided into two main groups according to 
their electrical resistivity: conductors and insulators. Conductors were 
loosely defined as having resistivities less than about 10~ 3 ohm cm, and this 
group of materials included all the metals. Insulators were considered to 
include substances with resistivities higher than 10 8 ohm cm and in this 
category were ebonite, glass, mica, etc. Materials having a resistivity between 
these two values were known, but it was often difficult to measure their 
resistivity accurately since different specimens of the same substance gave 
different results. Such materials are called semiconductors, and an explana- 
tion of their electrical properties will be given later in this chapter. The 
elements germanium and silicon are the two most used semiconductors, and 
the electrical properties of these materials are responsible for the operation of 
transistors and other semiconducting devices. 

The transistor is the most important of the solid-state devices. In a solid, 
the passage of current is determined by the binding forces which act between 
electrons and the positively charged atomic nuclei. Therefore, the emphasis 
in beginning this study of modern electronic theory will be placed upon an 
understanding of the physical properties of the atom. 



1-2 Simple Theory of the Hydrogen Atom 

Rutherford first proposed an atomic model of the type shown in Fig. 1-1. 
In this diagram, an electron is shown revolving in a circular orbit around a 
positively charged nucleus. In the case of hydrogen, it is known that the 
positive charge in the nucleus is numerically equal to the charge on the 



Circular electron orbit 

of unrestricted 
diameter (charge - e) 



Central, positively charged 
nucleus (charge +e) 



FIG. 1-1. Simple picture of the hydrogen atom. 




SEC. 1-2 SIMPLE THEORY OF THE HYDROGEN ATOM 5 

electron (e). Coulomb's law states that the force of attraction between these 
two charges is 



4-77£ r 



where r is the radius of the orbit in meters, and s is the permittivity of a 
vacuum. 

If the electron is revolving with velocity v in meters per second, the radial 
force necessary to restrain it in a circular orbit of radius r is mv 2 /r. Hence, 



(1-1) 



47T£ r 



Equation (1-1) is seen to be a result of the application of the classical laws 
of mechanics and electrostatics. It does not place any restriction on the 
radius of the electron orbit, and it is inadequate to explain why the electron 
remains in equilibrium in its path around the nucleus. Any change in the 
radius of the orbit will be accompanied by a change in the energy of the 
system [see Eq. (1-10)]. According to this theory, if the electron moves to a 
new orbit closer to the central nucleus, the energy of the system falls, and the 
excess energy is radiated out into space. This process will be repeated as 
time goes on and eventually the electron will fall into the nucleus. It is clear 
that Eq. (1-1) alone is incapable of explaining the motion of the electron 
around the nucleus. 

An additional discrepancy between this simple theory and experiment is 
that this model does not explain the mechanism of the emission and absorp- 
tion of radiation which takes place in a gas. Studies of the emission and 
absorption spectra of various gases show that certain frequencies appear 
which are characteristic of the gas. It is concluded, therefore, that definite 
energy levels are possible in the atom and that emission or absorption of 
radiation occurs when electrons move from one energy level to another. 
Classical theory is unable to explain this phenomenon since it does not 
indicate the presence of definite electron energy levels in the atom. 

1-3 Bohr's Theory of the Hydrogen Atom 

In 1913, Bohr suggested that these difficulties could be overcome by a direct 
application of Planck's quantum theory. The idea of a quantum of energy is 
fundamental to modern electronic theory since it is concerned with the 
emission and absorption of energy on an atomic scale. This was one of the 
first of many successful applications of quantum theory in the solution of 
atomic problems and led to the later and more exact theory of the atom. 

The quantum theory states that all energy is emitted and absorbed in the 
form of multiples of a unit called the quantum. Unlike the more familiar 



6 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

units of energy, the energy in each quantum is a function of frequency and is 
given by 

E = hv (1-2) 

where E is the energy of one quantum, in joules; h is Planck's constant, 
6.62 x 10 ~ 34 joule sec; and v is the frequency of radiation, in cycles per 
second. The amount of energy in a quantum is very small so that the emission 
and absorption of energy in experiments in ordinary mechanics are not 
affected. However, many atomic phenomena are governed by laws which 
may be deduced from the quantum theory. 

Bohr applied this theory to show that it was possible to restrict the orbital 
radii to certain discrete values. He imposed a quantum condition on the value 
of the angular momentum in an orbit and showed that this was necessary for 
the establishment of equations which fitted the experimental facts. When the 
magnitude of any quantity is restricted to multiples of a small unit (such as a 
quantum of energy), the quantity is said to be "quantized." The quantization 
carried out by Bohr [Eq. (1-5)] was later shown to be the restriction of angular 
momentum to multiples of a small unit. 

Bohr's model of the atom is based upon two assumptions. 

1. The radius of the orbit is limited to certain fixed values, and no 
emission or absorption of energy takes place while the electron remains 
in one of the permitted orbits. 

2. When the electron jumps from an orbit of one allowed radius to 
another, emission or absorption of a quantum of energy occurs. The 
frequency of radiation can be found from 

hv = E 2 - E 1 (1-3) 

where E 2 and E 1 are the energy levels of the two electron orbits. 

Assuming that the electron revolves in a circle around the nucleus, its 
angular momentum is 

mvr (1-4) 

The quantum condition imposed by Bohr on the angular momentum was that 

nh . 

mvr = x— (1-5) 

lm 

where n is a positive integer known as the principal quantum number. This 
number determines the energy of the orbit. 

Eliminating v from Eqs. (1-1) and (1-5) gives 

r n = 2" ( 1_6 ) 

where r„ is the radius of the «th orbit. 



SEC. 1-3 BOHR'S THEORY OF THE HYDROGEN ATOM 



Inserting the known values of the physical constants, //, £ , m, and e (see 
Appendix) in Eq. (1-6) and taking // = 1 



i\ = 0.529 x 10 " 10 meter 

= 0.529 A (Angstrom units) 



(1-7) 



The value of r l for the hydrogen atom is known as the Bohr radius, r B . It 
is of the same order as the effective radius of the atom found from kinetic 
theory. From Eq. (1-6), 



fi = r B 

7*4 



9r B 

\6r B etc. 



(1-8) 



In this way, Bohr suggested that the allowed radii that the electron orbits 
could take were in the ratio l 2 , 2 2 , 3 2 , 4 2 , . . .. This is shown in Fig. 1-2. 




n = 3 



FIG. 1-2. Bohr theory of the hydrogen atom showing the first 
three electron orbits. 



1-4 Energy of the Electron Orbits of the Hydrogen Atom 

The kinetic energy of an electron revolving in an orbit with velocity v is 
%mv 2 . From Eq. (1-1), 

«2 



K.E. = 



87T£ r n 



(1-9) 



8 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



The potential energy of this electron is the product of the electric potential 
at the point and the charge on the electron. 



P.E. 



47re r n 



(-«) = 



— e 



47r£ r 7 



(1-10) 



Note that the potential energy is negative and numerically is twice the 
value of the kinetic energy. In electrical systems, the potential energy at a 
point may be negative without any special significance being attached to the 
fact. Electrical potential is defined as the work done against the field in 
bringing a unit positive charge from infinity up to the point; the electrical 




Distance 
from nucleus 



Distance 
from nucleus 



Total energy of 
electron 



Position of 
nucleus 

FIG. 1-3. Plot of the total energy of an electron close to the 
hydrogen nucleus. 



potential at a point near a negative charge is, therefore, negative in sign. This 
definition means that the potential energy of the ionized atom is taken to be 
zero (see Fig. 1-3). 

The total energy of the electron in orbit of radius r n is 



K.E. + P.E. 

Substituting for r n from Eq. (1-6), 



877£ r n 



Total energy, E n = — 



me 



8n 2 h 2 e 2 
where E n is the energy of the nth orbital state (radius r n ). 



(1-11) 



(1-12) 



SEC. 



ENERGY OF THE ELECTRON ORBITS OF THE HYDROGEN ATOM 



The inner orbit where n — 1 has the lowest (i.e., the most negative) 
energy and is known as the ground state. In the hydrogen atom, the electron 
will remain in this state unless it is disturbed by outside influences. 

1-5 Transitions between the Orbits 

The simple case of hydrogen serves as a useful starting point towards under- 
standing the way in which the absorption and emission of energy take place 





Ionized state (electron free from the atom) 


- 


i 




-1- 






-2- 






-3- 


Excited state 














-4- 










-5- 










-6- 






n = 2to 




-7- 






n = l 


71= 1 to 

n=oo 


-8- 




n = l to 
n = 2 






-9- 










-10- 














A 


B 


C 


-11- 










-12- 










-13- 






Ground state 




-14- 











n = 4 
n = S 



n = 2 



n = l 



FIG. 1-4. Energy level representation for the Bohr theory of the 
hydrogen atom. 



in an atomic system. The single orbital electron of a hydrogen atom can be 
removed from the ground state n = 1 and brought into the state n = 2 if 
electromagnetic radiation of a certain frequency is incident on the atom. To 
move the electron from one orbit to another requires a quantum of energy 



10 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

equal to the energy difference of the two orbits. A quantum of electro- 
magnetic energy such as is absorbed by one atom in this process is referred to 
as a photon of radiation and the frequency of the photon may be calculated as 
before from 

hv = E 2 - E, (1-3) 

It is found that a photon of energy of this wavelength is absorbed by only 
one atom causing the electron to move from n — 1 to n = 2. In this sense, 
the photon of radiation acts as if it were concentrated at a point in apparent 
contradiction to the usual concept of wave motion. This particle-like nature 
of a photon is a concept which has been extended and developed to provide 
the basis for more advanced work on quantum mechanics which will be 
briefly discussed later. 

The atom is said to be excited when the electron is raised to a new orbit. 
If the electron is given sufficient energy to remove it completely from the 
influence of the nucleus, the atom is ionized, and the energy necessary to 
release an electron in this way is called the ionization energy, E h The ioniza- 
tion energy may be calculated in the case of hydrogen by considering the 
difference in the energy between orbits n = 1 and n = oo. The quantum 
number oo means that the electron is completely removed from the nucleus 
From Eq. (1-12), 

Ei = E x — Ei 

= -jn^ [1 _ II = »*L (i_i3) 

Sh 2 e 2 Loo lj Sh 2 e 2 K } 

Putting in the known values of the physical constants, 

E = 13.60 ev (electron volts) (1-14) 

where one electron volt is the energy which an electron acquires in moving 
through a potential difference of one volt and equals 1.6 x 10" 19 joule. 
Equation (1-14) is in agreement with the experimentally measured values. 

1-6 Energy Level Representation 

Figure 1-4 shows a representation of the energy levels of the first four orbits 
in the hydrogen atom as predicted by the Bohr theory. The energy of the 
electron states is plotted vertically with the positively increasing direction 
upwards and with zero energy at the top. The horizontal axis has no signifi- 
cance here, but in diagrams of this nature which will be used later, it will 
indicate distance in a material if the energy of the states varies for any reason 
through the substance. 

The ground state for which n = 1 is normally occupied. If radiation of 
frequency given by Eq. (1-3) is incident on the atom, the electron will move 
into orbit n = 2. This transition is indicated by the line A. After remaining 



SEC. 1-6 ENERGY LEVEL REPRESENTATION I I 

in the excited state for about 10" 8 second, the reverse transition B will occur, 
and energy in the form of a photon of frequency v will be emitted from the 
atom. The transition C indicates ionization of the atom. 

This method of indicating the energy levels of the various possible electron 
orbits is a most important one and will be used later not only for gases but 
also for solid-state materials. 

1-7 Limitations of the Bohr Theory 

The Bohr theory was the first step towards understanding the structure of 
the atom as it was able to explain the presence of certain frequencies in the 
emission and absorption spectrum of hydrogen which spectroscopists had 
already observed. The theory also predicted that other frequencies would be 
found in the spectrum corresponding to the transitions between various 
orbits as indicated by Eq. (1-3); these spectral lines were later detected. 
However, Bohr's theory of the atom was found to be limited to the hydrogen 
atom and to other atoms which contained only one electron, such as singly 
ionized helium and doubly ionized lithium. All attempts to apply the theory 
to many-electron atoms failed to predict the atomic energy levels which were 
known to exist. 

Between 1913 and 1925, many attempts were made by Bohr, Sommerfeld, 
and others to revise the simple theory so that it more accurately predicted the 
results of experiments. The principal modification was to assume that the 
electrons were traveling in elliptical orbits; extra quantum numbers were 
assigned to limit the value of the eccentricity of each orbit. These attempts, 
although partially successful, were completely superseded in 1925 by the 
wave-mechanical approach. 

In spite of these difficulties, the Bohr theory was an outstanding step 
forward in understanding energy levels, and the ideas which have been 
presented here are fundamental to an understanding of the mechanism of 
electrical conduction and electronic phenomena in both gases and solids. 

1-8 The Wave Nature of Matter 

The problem of a more accurate calculation of the energy levels in atoms 
was effectively solved between 1925 and 1930 as a result of a growing aware- 
ness of the duality between radiation and matter. The "particle-like" nature 
of a photon of radiation has been mentioned previously in connection with 
the absorption of a photon by an atom. The relatively successful theory of the 
atom due to Bohr, and other theories of the photoelectric effect, can only 
be explained by considering the photon to be localized over a small region. 
Thus the photon was considered to be of the nature of a particle, even though 
it was difficult to reconcile the conflicting properties of wave motion and 
those of a particle. 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. 



In 1922, de Broglie suggested that the converse of this effect also applies, 
that is, that a small particle (such as an electron) can be represented by a 
wave motion. He calculated that the wavelength, A, associated with a particle 
of mass m traveling at a velocity v is given by h/mv. Davisson and Germer, 
and also G. P. Thomson, performed experiments in 1927 to verify this and 
showed that there were diffraction phenomena associated with the passage of 
an electron beam into matter. Following their work, the wavelike properties 
of matter were generally accepted. The wavelength of the radiation varies 
with the velocity (or energy) of the beam. As an example, substitution of the 



Electron 



(a) 



"N 



■> Velocity v 



•s a .2 

£ M « 
« S & 



•S x s 

In 

!!« 

O <D +a 



^A/iA/v^ 



(b) 




Distance 



FIG. 1-5. Simplified picture of the solution of the wave equation 
for a moving electron. 

known values of the physical constants shows that A = 1.22 A for electrons of 
energy 100 ev. 

Between 1926 and 1930, Schrodinger, Dirac, Heisenberg, and others 
developed the theoretical implications of the wave properties of matter and 
produced the mathematical analysis known as quantum (or wave) mechanics. 
As one of the first steps in this theory, Schrodinger developed the wave 
equation which now bears his name and which is an important tool in 
the analysis of atomic systems. In its simplest form, Schrodinger's wave 
equation may be written as 

d 2 i/j &7T 2 m 

8x 2 + 



(E - U)i/> = 



(1-15) 



where E is the total energy, and U is the potential energy as a function of 



SEC. 1-8 THE WAVE NATURE OF MATTER 13 

distance. The solution of this equation gives the wave function, i/j, in the x 
direction. The normalized value of </» 2 represents the probability of finding 
the particle at a given point. 

A simple example of the solution of Schrodinger's wave equation for the 
case of an electron moving freely in space is given in Fig. 1-5. The classical 
picture of a particle moving from left to right with velocity v is shown in 
Fig. l-5(a). Part (b) of the figure shows the solution of the wave equation and 
gives a plot of the wave function, 0, versus distance. The electron is repre- 
sented by a group of waves as opposed to a continuous wave train, and this 
group of waves is often known as a "wave packet." Comparison of parts (a) 
and (b) of Fig. 1-5 shows that the solution of the wave equation spreads the 
effect of the electron over a region whereas the classical model limits the 
electron to a point. It is found that the best description of the position of the 
electron can be given by specifying the probability of finding the electron at 
any point. This probability is plotted as a function of distance in Fig. l-5(c). 
Although this probability reaches a maximum, it is not possible to locate the 
electron precisely since there is some probability of finding the electron at 
any point in the region covered by the wave function of part (b). 

When only one electron is concerned in a problem, the concept of the 
probability of finding it at a given position is a difficult one. However, the 
concept becomes more understandable when many particles are present since 
the probability factor is then the fraction of the total number of particles 
involved. In this case, the probability (which may be calculated directly 
from quantum mechanics) gives statistical information. In the analysis of 
atomic systems, such statistical information is of great importance in deter- 
mining the properties of the system since, even though the probability for a 
process may be small, the total number of particles may be so high that the 
number involved in the process is considerable. In addition, it was noted that 
the quantum mechanical solution for a particle in free space spreads the 
effect of the particle over a wider region than can be predicted by using 
classical mechanics (Fig. 1-5). It is also found that certain processes, although 
classically impossible, become statistically possible using quantum mechanics. 
In other words, using quantum mechanics, a finite (but perhaps small) 
probability may be predicted for some phenomena or processes that are 
completely impossible according to the classical laws of physics. Electrical 
conduction in semiconductors is based upon such processes. 

One of the most interesting and fundamental ideas that has been developed 
from this approach is the relationship known as Heisenberg's uncertainty 
principle. It is found that there is a limit to the accuracy with which we can 
determine the position and the momentum of a particle by measuring these 
quantities simultaneously. If x represents the position and/? the momentum 
of a particle, and Ax and Ap represent the accuracy with which these quantities 
can be specified, 

ApAx~h (1-16) 



14 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

This relationship introduces the concept of uncertainty or indeterminancy into 
atomic physics. It means that for experiments which are conducted on an 
atomic scale, it is not possible to assign a definite position and momentum to 
a particle at the same time. The more accurately one of these quantities is 
determined, the less accurately the other one is known. Because of the small 
size of Planck's constant, h, this effect is important only in atomic phenomena 
and not in large-scale experiments in the laboratory. In this book, Heisen- 
berg's uncertainty principle is used to determine the properties of the conduc- 
tion electrons of a metal in Sec. 2-1. It is of great significance in any experi- 
ment or analysis where the energy and position of small particles are con- 
cerned. 



1-9 Wave-mechanical Theory of Atomic Structure 

One of the first applications of quantum mechanics was to provide a solution 
to the problem of the structure of the atom. It was found that the electron 
energy states require four quantum numbers to describe them. 

1. The principal quantum number, n, is an integer 1, 2, 3, 4, . . . which 
mainly determines the energy of the electron state. It corresponds to the 
quantum number n in the Bohr theory, but the radius of the electron orbit can 
no longer be stated with accuracy since the probability of finding the electron 
is finite over a considerable region around the nucleus. The principal quan- 
tum number specifies the radial variation of this probability. 

2. The angular momentum quantum number, I, has a much smaller effect 
upon the total energy of the electron state. It determines the angular varia- 
tion of this probability. / can take the values 0, 1, 2, 3, ...,(«— 1). 

3. The magnetic quantum number, m, has, at most, only a small effect on 
the electron state energy. It is so named because it determines the component 
of angular momentum along a particular direction which is of importance 
when the atom is in a magnetic field. The values of m are restricted to 
/,(/- l),(/-2),...,-(/- 1),-/. 

4. The spin quantum number, s, was first suggested empirically by Uhlen- 
beck and Goudsmit in 1925. This number can only take the values +■£ or 
— J, and it was introduced to explain certain features in atomic spectra by 
assuming that the electron rotates on its axis about its own center of mass. 
Later work by Dirac showed that the quantum number s could be introduced 
as the solution to a particular type of wave equation. The two values ± \ 
denote that the electron can rotate in either direction with \ quantum unit of 
angular momentum. 

It is not possible to fit the idea of four quantum numbers into an accurate 
mechanical model of the atom. However, the mechanical model is often 
retained in mind to give some picture (however inaccurate) of the atom. In 



o o o o o o 
o o 



o o 



SEC. 1-9 WAVE-MECHANICAL THEORY OF ATOMIC STRUCTURE 15 

this case, n may be thought of as specifying the major axis of an elliptical 
orbit; / specifies the minor axis of this orbit; m determines the position of the 
orbit in space; and s determines the direction in which the electron is spinning 
on its own axis. 

The energy level of the orbit is mainly decided by the principal quantum 
number, n, and the other quantum numbers have a relatively small influence 
on the level. However, for many-electron atoms, there are now small differ- 
ences in energy among electrons having the same value of n, and this explains 
the presence of many frequencies in the absorption and emission spectra that 
the Bohr theory was unable to predict. 

Figure 1-6 shows diagrammatically the first 

ten states which can be computed from the _ (i- 

solution of the wave equation. The values of the \i, 

quantum numbers for each of these states can be 

found by applying the algebraic rules given at 

the beginning of this section. For example, with 

n = 1, / and m are restricted to zero. The spin n = i, z=o 

quantum number, s, can then take the two values 

± J, and so there are two states corresponding to 

n = 1. For n = 2, there are eight possible states, FIG ' ' ~ b : Diagrammatic rep- 
. . - reservation of the first ten 

two with / = and six with / = 1. energy states 

All electrons which have the same value of n 

are said to be in the same shell. A shell may be further divided into subshells 

where each subshell has a different value of /. Thus, in Fig. 1-6, there are 

two subshells in the n = 2 shell. For the hydrogen atom, the levels shown 

in the first and second shells are the first ten allowed states where the electron 

may be found. When the atom is not excited or ionized, the electron is in the 

n = 1 shell. Transitions between the various states correspond to known 

emission and absorption frequencies. 



1-10 The Atomic Table 

Unlike the Bohr theory, the quantum mechanical solution of the atom is not 
limited to hydrogen but can be applied to all atoms. It is possible to compute 
the electron energy states of the atoms successively from hydrogen up through 
the atomic table. The element that has two electrons in its atom is helium. 
Lithium, berylium, boron, and carbon follow with three, four, five, and six 
electrons respectively. By considering the solution of the Schrddinger wave 
equation, it is possible to build up an atomic table of the elements as in Table 
1-1. Here, the values of n and / are given for the first 36 atoms. The column 
labeled atomic number is the positive charge on the nucleus and is equal to 
the number of electrons outside the nucleus since the atom as a whole is 
electrically neutral. 



16 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



TABLE I -I. The atomic table for the first 36 elements. 



Atomic 


Element 


First 


Second 




77uW 




Fourth 


number 




shell 


shell 




j/re// 




shell 






n = 1 


n = 


= 2 




« = 3 




n = 4 


Z 


/ = 


I = 


/= 1 


/ = 


/= 1 


/ = 2 


/ = / = 1 


1 


H 


, 














2 


He 


2 














3 


Li 


2 


, 












4 


Be 


2 


2 












5 


B 


2 


2 


1 










6 


C 


2 


2 


2 










7 


N 


2 


2 


3 










8 


O 


2 


2 


4 










9 


F 


2 


2 


5 










10 


Ne 


2 


2 


6 










11 


Na 


2 


2 


6 


1 








12 


Mg 


2 


2 


6 


2 








13 


Al 


2 


2 


6 


2 


1 






14 


Si 


2 


2 


6 


2 


2 






15 


P 


2 


2 


6 


2 


3 






16 


S 


2 


2 


6 


2 


4 






17 


CI 


2 


2 


6 


2 


5 






18 


A 


2 


2 


6 


2 


6 






19 


K 


2 


2 


6 


2 


6 




1 


20 


Ca 


2 


2 


6 


2 


6 




2 


21 


Sc 


2 


2 


6 


2 


6 


1 


2 


22 


Ti 


2 


2 


6 


2 


6 


2 


2 


23 


V 


2 


2 


6 


2 


6 


3 


2 


24 


Cr 


2 


2 


6 


2 


6 


5 


1 


25 


Mn 


2 


2 


6 


2 


6 


5 


2 


26 


Fe 


2 


2 


6 


2 


6 


6 


2 


27 


Co 


2 


2 


6 


2 


6 


7 


2 


28 


Ni 


2 


2 


6 


2 


6 


8 


2 


29 


Cu 


2 


2 


6 


2 


6 


10 


1 


30 


Zn 


2 


2 


6 


2 


6 


10 


2 


31 


Ga 


2 


2 


6 


2 


6 


10 


2 1 


32 


Ge 


2 


2 


6 


2 


6 


10 


2 2 


33 


As 


2 


2 


6 


2 


6 


10 


2 3 


34 


Se 


2 


2 


6 


2 


6 


10 


2 4 


35 


Br 


2 


2 


6 


2 


6 


10 


2 5 


36 


Kr 


2 


2 


6 


2 


6 


10 


2 6 



SEC. I — 10 THE ATOMIC TABLE 17 

There are two rules which govern how the atomic table is built up. First, 
in any atom, the electrons will normally occupy the states having the lowest 
energy. Secondly, no two electrons in an atom can have the same set of four 
quantum numbers. This second factor was first enunciated by Pauli in 1925 
and is known as the exclusion principle. 

The first ten elements, which have electrons only in the first or second 
shells, are arranged in the table in the regular manner shown. Only values of 
n and / are given and the numbers in each column show how many electrons 
have that value of / (although they will have different values of m and s). 
From the table it is seen that both levels in the first shell are occupied for 
helium. Lithium has the first shell filled and an additional electron in an 
n = 2, / = state. Berylium has the same filled first shell and two electrons 
in the n = 2, / = subshell. This sequence continues up the table until 
element number ten, neon, is found to have a completely filled second shell. 

The third and later shells do not possess the simple regularity shown in the 
table for the first two shells. It is seen that argon (Z = 18) has a closed / = 1 
subshell but has no electrons corresponding to / = 2. Other irregularities 
occur in the fourth shell. In all cases, however, the lowest empty energy state 
is filled by the extra electron of the succeeding element. 

Inspection of the atomic table shows that the substances having the 
maximum number of electrons in a shell or subshell are helium (n = 1), neon 
(n = 2), argon (n = 3), and krypton (n = 4). These substances are said to 
have closed electron shells (or subshells) and they are chemically inert. 
Their behavior gives a clue to the way in which elements combine together. 
The closed outer shell indicates chemical stability, and the number of electrons 
which an electron has missing from a closed shell, or in excess of a closed shell, 
determines the chemical properties of the material. 

I — 1 1 Chemical Valence and Atomic Binding 

The number of electrons which an element has in excess of the nearest closed 
shell is known as a positive valence. The number of electrons below the 
nearest closed shell is a negative valence. When two substances combine, 
they do so in such a manner that the combination approaches chemical 
stability. In a simple way, it may be said that compounds will be built up 
between elements which complement each other to form closed shells. How- 
ever, when substances combine, the electron energy levels of both substances 
are altered, and the energies of the electrons in the outer shells are the most 
affected. So the simple picture of valence is only approximate, and recourse 
must be made to the mathematical approach of quantum mechanics to 
determine the exchange of electrons between the atoms in the general case. 
The elements fluorine (Z = 9) and chlorine (Z = 17) have one electron 
missing from the closed shell n = 2 and n = 3 respectively (see Table 1-1) 
and so they have similar chemical properties. When they combine in chemical 



18 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP, I 

compounds, they try to acquire an electron from another substance to com- 
plete the electron shell. Upon consulting the atomic table, it can be seen that 
lithium (Z = 3) and sodium (Z = 11), etc., are elements which have an 
electron in excess of the closed shell that can be used in this process. This 
class of binding between two atoms is known as ionic since one atom of the 
pair gains an electron and becomes negative and the other one loses an 
electron and becomes positive. The resulting bond between the two elements 
is due to the coulomb attractive force between the positive and negative ion. 
The two semiconducting elements germanium and silicon take part in a 
binding process which is different in nature from the ionic mechanism 
described above. These elements have a valence of ±4 meaning that four 
electrons are required to close the shell and also that there are four electrons 
present in the outer shell (or subshell in the case of germanium). The element 
carbon (Z = 6) also has this property, and all three substances take part in 
covalent binding and, in their solid phases, can exist in the same structural 
form. This form is characterized by each atom having four near neighbors. 
The four outer (valence) electrons of each atom require four more electrons 
to produce a closed shell (or subshell). These four electrons are obtained 
from the four neighbors. Every valence electron, therefore, may be considered 
to be shared between two atoms : its parent and one of the neighbors. Thus, 
every atom shares the eight valence electrons it requires with its neighbors. 
The way in which atoms of any one of these substances combine together to 
form a solid crystal is important in semiconductor theory, and covalent 
binding will be described in the next section in terms of the crystal structure 
of germanium and silicon. 

1-12 Crystals 

A crystal is a solid piece of material in which atoms are arranged in a regular 
pattern. The structure of the crystal is determined by the way in which the 
individual atoms are chemically bound to one another. A perfect crystal 
consists of the repetition of a unit pattern or cell of atoms where all the units 
are oriented in the same direction with no irregularities. The nearest approach 
to this, in practice, is a single crystal where all the unit cells are arranged in 
the same direction but some structural discontinuities (defects) are present in 
the material. These discontinuities may be associated with the presence of a 
foreign atom or may result from temperature variations when the crystal was 
being formed. The imperfections in a crystal are extremely important in de- 
termining the mechanical and electrical properties of the material. Semi- 
conducting devices are constructed from single crystals which are grown with 
deliberate additions of chemical impurities (imperfections). Single crystals 
are rarely found in nature but they may be produced using careful crystal 
growing techniques. Single crystals of germanium or silicon with minute 
amounts of impurities are used in the manufacture of transistors, and these 



SEC. 1-12 



CRYSTALS 



19 



crystals possess particular electrical properties which are discussed later in 
this chapter. In nature, most crystalline substances occur in polycrystalline 
form where different crystal orientation is found in different parts of the 
material. Such crystals are generally unsuitable for semiconducting devices. 



fa) 








\ 6 /^ 

9 * ' 

/V 


Xl2 

10X 






~A-~- 



(c) 



FIG. 1-7. Three types of crystal unit cells. 



The germanium and silicon crystal structure is similar to the diamond 
structure of carbon in that atoms are arranged with covalent bonds to their 
four nearest neighbors. The unit cell of this type of structure is not simple to 
understand at first sight as a glance at Fig. 1-8 will show. Before discussing 
the diamond crystal, consider the three unit cubic cells drawn in Fig. 1-7. 
In Fig. l-7(a), the simple cell has atoms at each corner. By considering a 




FIG. 1-8. Crystal structure of diamond, silicon, and germanium. 
A model of the diamond lattice. Each sphere represents a 
nucleus surrounded by its inner shells. The spokes represent 
covalent bonds. 



number of these cells placed together, it will be seen that every atom has six 
nearest neighbors spaced apart by the side of the cube. Figure 1— 7(b) shows 
a more complex arrangement which is known as a body-centered cubic cell. 
Here there is one additional atom in the middle of the cube, and this atom 
has eight nearest neighbors. Finally, the face-centered cube is shown in Fig. 



20 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

l-7(c). This structure will be shown to be a component of the diamond 
crystal. For the same size of unit cell, there are more atoms in the face- 
centered cube than in the body-centered cell and every atom in the face- 
centered cell will be seen to have twelve nearest neighbors. 

The atoms on the diamond structure in Fig. 1-8 have been numbered for 
identification purposes. Atoms 1 through 8 are at the corners of a face- 
centered cube. Atoms 9 through 14 are on the six faces of the cube. Four 
atoms, numbers 15, 16, 17, and 18 remain. These are the atoms of another, 
similar, face-centered cubic cell which is spaced one quarter of the cube 
diagonal away from the first unit cell (the diagonal is formed by a line from 1 
to 7). Atoms 15, 16, 17, and 18 are the only atoms of this new cell which are 
contained within the limits of the first cube, and they correspond to numbers 
1, 13, 9, and 12 of the first cell. The remaining ten atoms are outside the first 
cell and are not shown on the diagram. As an aid in picturing the three- 
dimensional structure, consider the five horizontal layers of atoms in the 
cube. The bottom layer consists of atoms 1, 2, 3, 4, and 13 on the base of the 
cube. The second layer which is one quarter of the way up the cube contains 
the added atoms numbers 15 and 16. Halfway up the cell are the four original 
face atoms 9, 10, 11, and 12, and three quarters of the way up are added atoms 
17 and 18. The top face contains numbers 5, 6, 7, 8, and 14. The distance 
between atoms 3 and 7 is known as the lattice constant for the crystal. The 
lattice constants for the three materials are: diamond, 3.56 A; silicon, 5.43 A; 
germanium, 5.66 A. 

Consider now atom 17 in relation to its nearest neighbors, atoms 9, 10, 6, 
and 14 (shown shaded). These five atoms share electrons in covalent bonds. 
Any other atom in the cell can be shown to have four nearest neighbors 
although not all of these neighbors are on the diagram. 



1-13 Electron Energy Levels in Crystals: Band Theory 

The electron energy level diagrams that have been given for single, free atoms 
will no longer apply to the same atom in a crystal. The energy levels of the 
inner electrons will be changed only a small amount since the inner (closed 
shell) electrons are not affected very much by the presence of neighboring 
atoms. However, the levels of the outer (valence) electrons are greatly 
affected since these are the electrons which are participating in chemical bonds. 
Once more, recourse must be made to quantum mechanics in order to de- 
termine the new energy levels since classical theory is inadequate to explain 
the binding process. 

There is a simple analogy which may be drawn between the energy of an 
electron in an outer orbit and the frequency of a tuned LC circuit. In an 
atom that is isolated from its neighbors, an electron has been shown to possess 
a certain discrete energy level. Similarly, an isolated tuned circuit possesses a 



SEC. 1-13 



ELECTRON ENERGY LEVELS IN CRYSTALS: BAND THEORY 



21 



single resonant frequency of value 

fo = 



iWlc 



when the circuit resistance is neglected. The analogy between the energy of 
the electron state and the frequency of the resonant circuit may be extended 
by considering the effects of coupling in both cases. Figure 1-9 illustrates the 
effect of coupling several identical tuned circuits together. A sharply tuned 
resonance curve is obtained for the isolated tuned circuit of Fig. l-9(a). 
When two similar circuits are coupled together, a resonance curve having two 
peaks as in Fig. 1— 9(b) is obtained. The introduction of further coupled 
circuits results in the broad resonance curve as shown in Fig. l-9(c). The 
width of the resonance curve depends only on the degree of coupling between 
the circuits and is independent of the number of circuits which are coupled 
together. 



Coupled i 



& 



JL S\ 



fo 


fo 


Frequency — ► 


Frequency 


(a) 


(b) 



Coupled 




J~X 

fo 

Frequency — ► 

(c) 



FIG. 1-9. The effect of coupling upon the resonance curve of 
tuned circuits. 



Coupling between the outer electrons of neighboring atoms in a crystal 
has an effect upon the electron energy levels that is analogous to the broaden- 
ing of the resonance curve of coupled tuned circuits. It is found that there is 
now a band of energies instead of a discrete energy level as in the single atom. 
The energy of the outer electrons of any particular atom in the crystal must 
lie within this band, and it is not possible to specify an individual level any 
more than it is possible to specify the individual resonant frequency of one of 
a large number of coupled circuits. The width of the energy band depends 
upon the coupling between the outer electrons of the atom, that is, upon the 
distance between atoms in the crystal. If the interatomic distance is decreased, 
the coupling increases, and the band width increases as shown in Fig. 1-10. 
This is a theoretical curve as it is not possible to vary continuously the 



22 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



interatomic distance in a crystal, but it is introduced here as an aid to the 
understanding of the band theory of electron energies in crystals. 

In carbon, there are two electrons in the n = 1 shell and four in the n = 2 
shell. However, there are eight energy states in the complete n = 2 shell. In 
the isolated, free atom which is shown on the right-hand side of Fig. 1-10, 
six electron states have been shown filled, and the remaining four states in the 
n = 2 shell are empty. As the interatomic distance is decreased, the discrete 
levels broaden out into bands, and eventually the bands corresponding to the 
two subshells of n = 2 overlap as shown in the center of the diagram. As the 
interatomic distance is further decreased, a split occurs between the upper 



• Occupied state 
o States per atom 




Interatomic distance 



FIG. 1-10. The energy bands in carbon as a function of the inter- 
atomic distance (theoretical and not to scale). 

four states and the lower four states, and at the point marked "diamond 
crystal," the two bands are widely separated. The electron level correspond- 
ing to n — 1 hardly changes as the interatomic distance is decreased. 

Figure 1-11 shows an energy band diagram for a diamond crystal plotted 
in the same way as the energy level diagrams that have been discussed pre- 
viously. This diagram is a section through the curve of Fig. 1-10 on the line 
marked "diamond crystal" where the energy band for n = 2 has split into 
two parts. The upper part is known as the conduction band, and the lower 
part is the valence band. In between the two, there is a region where no 
electron energy levels are permitted, and this region is known as the forbidden 
gap. At the bottom of the diagram is shown the very narrow band corre- 
sponding to n = 1. All the bands are shown bounded by straight horizontal 
lines since the crystal is assumed to be uniform. 



SEC. 1-13 



ELECTRON ENERGY LEVELS IN CRYSTALS: BAND THEORY 



23 



An atom of carbon has six electrons and so the lowest six levels per atom 
will be filled. Thus in Fig. 1-11, the two levels corresponding to n = 1 and 
the four levels per atom in the valence band are filled, leaving the conduction 
band completely empty. In practice, this is only approximately correct since 
we have neglected the effect of any heat energy which will be imparted to the 
electrons in the crystal at any given temperature. If this energy is so small that 
it has no effect on the distribution of electrons in the levels, conditions are 
exactly as shown in Fig. 1-11. 

The reason for the names of the valence and conduction bands now be- 
comes clear. The four outer electrons per atom, which are in the valence band, 
are the four electrons that determine the valence of carbon and that are 
participating in covalent bonds. These electrons, which are being shared with 
neighboring atoms, are closely bound to the nuclei and so their energy is lower 



Conduction band 



r 

Forbidden gap 6 ev 




ZZZZZZZZZZZZZZZZZZZZZZZZZZZ 



Four levels per atom 
(empty) 



Four levels per atom 
(filled) 



Two levels per atom 
(filled) 



Distance — ► 
FIG. I -I I. Energy band diagram for a diamond crystal. 



(more negative) than the energy of the corresponding states of free atoms 
(Fig. 1-10). If one electron is removed from a covalent bond in a diamond 
crystal, it will have acquired sufficient energy to move into an energy state 
above that of the bound electron. This may be represented on Fig. 1-11 as 
the passage of an electron from the valence band to the conduction band. 
The minimum energy that must be imparted to the electron to remove it from 
the covalent bond is the energy difference between the top of the valence 
band and the bottom of the conduction band, that is, the width of the for- 
bidden gap (6 ev in a diamond crystal). When the electron is in the conduc- 
tion band (removed from a covalent bond) it can move from atom to atom 
since the conduction band is empty. An electron, once removed from the 
valence band to the conduction band, is available to move in any electric 
field that may be applied across the material. 



24 

1-14 Insulators 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



Figure 1-11, although applying specifically to the diamond crystal, is also a 
typical energy band diagram of an insulator. The requirements for a crys- 
talline substance to be an insulator at room temperature are that the valence 
band must be full, the conduction band must be normally empty, and the 
forbidden gap must be greater than about one electron volt. Under these 
conditions, only a minute fraction of the total number of electrons in the 
material can gain enough energy to cross the forbidden gap and reach the 
conduction band. It will be shown later that the average energy possessed by 
an electron at room temperature (300° K) is only 26 millielectron volts 



o = o = o - o 

II II II 

o = o = o 

II II II 

o 



o 

II 



o = v o = o 

ii ii V ii ii 

o=o=o=o 





Conduction band 


1 

6ev 

4 l 


Forbidden 
gap 


1 


t 


////// Valence band <//%//; 







Electron moving 
from valence to 
conduction band 



(a) 



(b) 



Distance 



FIG. 1-12. Simplified picture of an electron breaking from a 
covalent bond in a diamond crystal. 



whereas it requires 6 ev of energy to move an electron from the valence to the 
conduction band in diamond. When an electric field is applied across a 
crystal, there is a flow of electrons in the conduction band. Diamond, with 
very few electrons in the conduction band, is therefore an extremely good in- 
sulator. If the temperature of the material is increased, the average energy of 
the electrons rises, and the number of electrons with enough energy to cross 
the gap also increases. So the conductivity of a good insulator, such as 
diamond, increases with temperature. 

Figure 1-1 2(a) shows a two-dimensional, simplified picture of a diamond 
crystal where the electrons that are held in covalent bonds are indicated by 
short lines. The arrow on this diagram indicates that one electron is about to 
break away from its bond, and it will then be free to move about in the crystal. 
In the energy band diagram of Fig. 1-1 2(b), this process is indicated by the 
arrow from the valence band to the conduction band. Once in the conduction 
band, the electron is free from the covalent binding force and can move 
freely in the crystal. In Fig. 1-1 2(a), the two inner electrons are not shown 
since they play no part in the conduction process; the energy band corre- 
sponding to these two electrons also has been omitted in Fig. 1-1 2(b). 



SEC. 1-15 CONDUCTORS 

1-15 Conductors 



25 



In a conductor, the outer electrons are only loosely bound to the parent 
nucleus, and the permissible electron levels are as indicated in Fig. 1-13. 
In this diagram, only the highest energy band is shown. This may be a single 
band, or it may be the result of the overlapping of two energy bands. How- 
ever it is formed, it has more permitted electron levels than there are available 
electrons, and so there are empty levels immediately above the highest filled 
levels. Consider an electron which has an energy corresponding to the top 
of the filled portion of the band. If this electron gains a small amount of 
energy from any source, it can move up into the empty part of the band. It is 
then free to move about in the crystal because the energy states at this level 




Total band width 
Band only partially full 



Distance — > 
FIG. 1-13. Band structure of a metal. 

are not occupied. Since there is no forbidden gap in the energy band just 
above the filled levels, electrons are always free to move through the crystal 
under the influence of an electric field. A material of this type is a conductor. 

1-16 Semiconductors 



A semiconductor has the energy band structure of an insulator. However, an 
insulator has a forbidden gap which is so wide that very few electrons can 
cross it at room temperature, whereas a semiconductor possesses a narrow 
gap which allows a considerable amount of conduction at room temperature. 
Germanium and silicon are the two most used semiconducting elements, and 
the widths of the forbidden gap in these materials are 0.76 and 1.1 ev respec- 
tively. These two materials have a resistivity intermediate between a conduc- 
tor (such as copper) and an insulator (such as diamond). 

When the conductivity in a crystalline semiconductor is due solely to the 
breaking of covalent bonds, the substance is said to be an intrinsic semi- 
conductor. The intrinsic effect is often masked by conduction because of the 
presence of impurity atoms in the crystal. As explained later in this chapter, 



26 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



certain atoms introduced into the crystal produce localized energy states 
which are in the forbidden energy gap, and these may result in electrical 
conduction. A material where this latter mechanism is dominant is known as 
an extrinsic semiconductor. 



1-17 Electrons and Positive Holes 

When an electron moves from the valence band into the conduction band of a 
semiconductor, it leaves behind an unfilled electron state. The absence of an 
electron in the valence band is called a positive hole. This expressive name 
indicates that a negatively charged electron is missing from the otherwise full 
band. There are several effects in the crystal that can be attributed to the 
absence of an electron in the valence band. The positive hole is, therefore, 



Direction of electric field 




- Electrons 

+ Positive holes 



Distance — ► 
FIG. 1-14. Electrical conduction by electrons and positive holes. 

endowed with properties that lead to these effects occurring in the crystal. 
For convenience, the positive hole is regarded as the active particle in the 
valence band in the same way that the electron is considered to be active in 
the conduction band. As shown later, this leads to the concept that conduc- 
tion in semiconductors is caused not only by the motion of electrons in the 
conduction band but also by positive holes in the valence band. 

Before proceeding further, it should be stated that the properties of an 
electron in a crystal are different from those of a free electron. Forces of 
attraction and repulsion act upon an electron in a crystal because of the close 
proximity of the atomic nuclei and of other electrons. In addition, an 
electron in the conduction band is subjected to different conditions from an 
electron in the valence band. Thus, the properties assigned to a positive hole 
(such as mass, mobility, etc.) will not be the same as those of an electron in the 
conduction band. 



SEC. 1-17 ELECTRONS AND POSITIVE HOLES 27 

The mechanism of positive hole conduction can be seen by reference to 
Fig. 1-14. At 0° K, the valence band is completely full, and the conduction 
band is empty because no thermal energy is available in the crystal. At any 
other temperature, as shown in Fig. 1-14, there will be a few electrons which 
gain enough energy to traverse the forbidden gap to the conduction band 
and leave positive holes in the valence band. Four such electrons and positive 
holes are shown. If an electric field, S, is applied across the material such 
that the left-hand side is positive, electrons in the conduction band, being free 
to move, will travel from right to left causing an electronic current to flow. 

If every electron energy state in the valence band were occupied, there 
would be no possibility of electrical conduction taking place in this band 
since a strong electron bond would have to be broken to allow electron 
motion. However, the absence of four electrons provides four energy states 
into which neighboring electrons can move under the influence of the electric 

Direction of electric field — ► Direction of electric field — ► 

0=0=0=0 0=0=0=0 

11 11 11 11 11 11 11 11 

0=0=0=0 0=0=0=0 

11 11 11 11 11 11 11 11 

0=0=0=0 0=0=0=0 

II lv —II II II II I II 

o = o\=o = o o = o=/o = o 



t \ Positive hole v Positive hole 

(a) (b) 

FIG. 1-15. Apparent motion of positive hole in electric field. 

field. These electrons require little extra energy to make this transition since 
the state into which they are moving is in the same energy band. 

The motion of a positive hole in a two-dimensional simplified version of 
a semiconductor crystal is shown in Fig. 1-15. The electric field is in the 
same direction as in the previous figure. One positive hole is shown in Fig. 
1-1 5(a), and the electric field aids another electron to move into the hole. 
Part (b) of the figure shows the crystal after the event has taken place. The 
positive hole has apparently moved in the direction of the electric field, and 
this motion constitutes a passage of current. This positive hole current takes 
place exclusively in the valence band since only the closely bound covalent 
electron energy states are involved. There is also an electron current which 
occurs in the conduction band and is not shown in Fig. 1-15. This is the 
result of the drift in the electric field of the electron originally excited into the 
conduction band. 



28 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

To distinguish between the two forms of current which flow in a semi- 
conductor it is usual to refer to the motion of electrons in the conduction 
band and to that of positive holes in the valence band. Yet, it should be 
remembered that electrons are flowing in both cases, and it is only a matter of 
convenience to label one a "positive hole" current. 

1-18 Mobility 

When charged particles in a crystal move under the influence of an electric 
field, they attain a velocity which is proportional to the value of the field. 
Thus, 

v = \l£ (1-17) 

where v is the drift velocity in cm sec" \ and $ is the electric field in volts cm" 1 . 
The quantity /jl is known as the mobility of the particles and is measured in 
cm 2 volt" 1 sec" 1 . 

From Eq. (1-17), electrons apparently drift in a given electric field with a 
constant velocity. However, this velocity is an average value, and the elec- 
trons do not, in fact, move across the crystal at a uniform speed. Each 
electron accelerates in the field and, after traveling a short distance, suffers a 
collision. At each collision, the electron loses most of the velocity it has 
gained, and then it starts to accelerate once more under the influence of the 
field. This process is repeated, and when all the mobile electrons are con- 
sidered, the average velocity is given by Eq. (1-17). 

The value of the mobility in semiconductors depends upon temperature, 
purity of the material, and on whether electron or positive hole motion is 
taking place. It is usual to assign the symbol \x n to the mobility of electrons 
in the conduction band and \l v to positive holes in the valence band. For 
silicon and germanium, \x n is higher than /x p because electrons in the conduc- 
tion band are more free of the attractive forces exerted by the nuclei than are 
positive holes which move only as a result of the slower drift of electrons in 
covalent bonds. 

TABLE 1-2. Properties of germanium and silicon. 

Property Germanium Silicon Units 



Atomic number 






32 


14 




Atomic weight 






72.6 


28.1 




Density 






5.32 


2.40 


gem -3 


Relative permittivity 






16 


12 




Gap energy 






0.72 


1.1 


ev 


tit at 300° K 






2.5 x 10 13 


-1.6 x 10 10 


cm -3 


Intrinsic resistivity at 300° K 




47 


-230,000 


ohm cm 


Mobility /x n at 300° K 






3900 


1500 


cm 2 volt -1 sec -1 


Mobility /*, at 300° K 






1900 


500 


cm 2 volt -1 sec" 1 


Diffusion constant D n 


at 300° 


K 


99 


39 


cm 2 sec -1 


Diffusion constant D p 


at 300° 


K 


49 


13 


cm 2 sec ~ 1 



SEC. 1-18 MOBILITY 29 

The values of mobility in Table 1-2 apply to very pure semiconducting 
material. This is known as intrinsic material. It will be shown in Sec. 1-19 
that the resistivity of a semiconductor is decreased by the presence of a very 
small amount of certain types of impurity atoms. If these impurities are 
present, the mobility also decreases below its intrinsic value. This is due to the 
increased number of collisions which a charge carrier makes with the added 
impurity atoms. Curves showing the variation in mobility of electrons and 
holes in germanium and silicon as a function of the density of added impurity 
atoms are given in Sec. 1-19. 

Consider a semiconductor having an electron density in the conduction 
band given by n per unit volume. The charge which is available for conduction 
per unit volume is ne and this charge will move with a velocity v in field S. 
The current density is given by 

J n = nev (1-18) 

and the conductivity of the material due to electrons in the conduction band 
only is denoted by a n where 

J n nev ne^ n S ,. im 

where the subscript n applies to effects associated with electrons in the 
conduction band. 

If p is the positive hole density in the valence band and fi p is the positive 
hole mobility, it can be shown by similar reasoning that 

o p = pe[i v (1-20) 

and the total conductivity is 

a = °p + <*n = e(pfi p + nfi n ) (1-21) 

For an intrinsic semiconductor, such as pure germanium or silicon, the 
electrons in the conduction band have all been removed from the previously 
full valence band and so 

p = n = n { (1-22) 

where n t is the intrinsic density of electrons or holes in the material and so 

a = n { e{\i p + ix n ) (1-23) 

1-19 p- and n-type Semiconductors 

In the previous discussion, it was noted that small amounts of certain im- 
purities greatly affect the conductivity of an otherwise pure semiconducting 
crystal. This will now be explained by reference to the two separate con- 
tributions which electron and positive hole motion make to the total con- 
ductivity of the material. 



30 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. 



Figure 1-1 6(a) shows the two-dimensional representation of a crystal 
lattice containing only atoms of an element which has four outer electrons 
available for covalent binding. Substances of this type (e.g., silicon and 
germanium) are termed group 4 materials and will form a diamond-type 



© = © = ©=© 
ii ii ii ii 

©=©=©=© 



© = © = ©=© 
ii ii ii ii 

© = © = © = © 

(a) 



Conduction band 
(nearly empty) 



Forbidden gap 




(b) 



Distance 



FIG. 1-16. (a) Two-dimensional picture of the crystal structure of 
a group 4 material, (b) Energy bands of the group 4 crystal. 

crystal structure. The energy bands for this crystal are given in Fig. 1-1 6(b). 
Now assume that a few atoms of a substance having five outer electrons are 
introduced into the crystal. This is a group 5 substance such as phosphorus, 
arsenic, or antimony (see Table 1-1), and one atom of this new material is 



© = © = © = <&_. 
ii 

© 



© 
ii 

© 

(a) 



© = © = © 
II II II 

© = © = © 
II II II 

© = © = © 



Extra" 
electron 



(b) 



Conduction band 




Energy state 
of "extra" 
electron 



Distance 



FIG. 1-17. Addition of one atom of group 5 material to a group 
4 crystal. 



shown in the crystal lattice in Fig. 1-17. Four of the five outer electrons of the 
group 5 atom will fit into the covalent bonds and will be shared with the four 
neighboring atoms. The fifth outer electron, however, does not take part in 
this binding process and is comparatively free to move. It is subject to a force 



SEC. 1-19 p- AND n-TYPE SEMICONDUCTORS 31 

of attraction towards its parent nucleus, but this force is much smaller than 
the covalent binding forces represented by energies in the valence band. 
The energy state corresponding to this fifth electron is shown just below the 
conduction band in Fig. 1-17 and is in the forbidden gap of the pure group 4 
substance. 

A rough calculation of the force of attraction between the fifth outer 
electron and its parent atom can be made by referring back to the simple 
Bohr theory of the atom. If the fifth electron were not present, the remainder 
of the atom would have a net charge of + e. This may be considered as the 
nuclear charge of a new, Bohr-like atom with one electron in orbit around it. 
However, Bohr's analysis, as exemplified by Eq. (1-12), was applied to the 
case where the electron was moving in free space, whereas here the electron 
is in a crystalline material of relative permittivity, e r , greater than unity. 
Equation (1-12) is, therefore, modified to 

E ' n = ~Sn 2 h 2 e 2 e 2 (1_24) 

where E' n is the energy of the electron in the orbit number n. From Eqs. (1-13) 
and (1-14), the energy of the ground state of hydrogen (n = 1) is 



me 



Ei = ~%iki = 13 - 60ev (1_25) 



Thus, for the electron in the crystal 



t?> E i 13 - 60 n ™ 

In a germanium crystal, e r = 16 and so 

E[ = 1|^ = 0.053 ev (1-27) 

It is necessary to justify the assumption that the electron is far enough 
away from its parent atom that it is surrounded by germanium atoms. From 
Eq. (1-6), the radius of the first orbit becomes 

r'i = ^ = V» (1-28) 

nme 

For germanium, 

r \ = \6r B = 8.5 A (1-29) 

This value for the radius of the orbit of the fifth outer electron is greater than 
the lattice constant of germanium (5.66 A) and so the electron orbit is wide 
enough to carry it well away from its parent atom and into the germanium 
crystal (see Fig. 1-17). 



32 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



TABLE 1-3. Common donor and acceptor elements and 
their energy levels in germanium and silicon. 



Element 


Type of 


Semiconductor 


Position 




impurity 


Ge 


Si 




Phosphorous (P) 


Donor 


0.012 


0.044 


In ev below the 


Arsenic (As) 


Donor 


0.013 


0.049 


bottom of the 


Antimony (Sb) 


Donor 


0.010 


0.039 


conduction band 


Boron (B) 


Acceptor 


0.010 


0.045 


In ev above the 


Aluminium (Al) 


Acceptor 


0.010 


0.057 


top of the 


Gallium (Ga) 


Acceptor 


0.011 


0.065 


valence band 


Indium (In) 


Acceptor 


0.011 


0.160 





The Bohr theory presents a very simple picture of the atom, and the 
accuracy of the calculation is poor. Measurements have shown that the energy 
level of the fifth electron of a group 5 atom in a germanium crystal is about 



Conduction band 




Electron free to move 
in conduction band 



Positively ionized donor 
state (immobile) 



Distance 



FIG. 1-18. Representation on an energy band diagram of an 
electron being separated from a donor state. 

0.01 ev. For silicon (e r — 12), the measured values are commonly about 
0.04 ev. Slightly different values are found for different added elements (see 
"donor" in Table 1-3). 

Since it requires such a small energy to remove the electron from the 
group 5 atom, this energy can be supplied from thermal agitation in the crystal. 
The electron is usually separated from its parent atom at room temperatures 
and Fig. 1-18 shows how this process is represented on the energy band 
diagram. The electron has been removed from the parent atom leaving 
behind a positively ionized (unfilled) electron bond, and it is free to move 
under the influence of an electric field. However, there is no corresponding 
positive hole in the valence band, and so only the electron motion in the 
conduction band will contribute towards the conductivity. The extra energy 
state associated with the group 5 material is known as a donor state. When 



SEC. I- 



p. AND n-TYPE SEMICONDUCTORS 



33 



the electron is removed, the donor state becomes positively ionized, but the 
positive charge is fixed in the crystal lattice and cannot move under the 
influence of an electric field. 

In intrinsic semiconductors where no impurity atoms are present, the 
number of electrons in the conduction band at room temperatures is a 
minute fraction of the total number of electrons in the crystal. Since each 
group 5 atom donates one electron to the conduction band, adding a very 
small fraction of impurity atoms to the number of group 4 atoms in the crystal 
increases the conductivity appreciably. As an example, a ratio of 1 atom of 
phosphorus to 10 8 atoms of germanium alters the conductivity of germanium 
at 27° C by a ratio of 12:1. 

When the conductivity of a semiconductor is determined largely by the 
added group 5 atoms, the material is known as n (for negative) type. It 
must be emphasized that the material as a whole is neutral since the added 
group 5 atoms have as many positive charges in the nucleus as they have 



Conduction band 




Additional energy state 
of group 3 atom 



Distance 



FIG. 1-19. Position of the unfilled energy state of group 3 atom. 



negative electrons. The term n-type is used to indicate that the major part of 
the current is carried by electrons in the conduction band. Thus, electrons 
are known as the majority carriers, and positive holes are the minority carriers 
in the fl-type material. 

An atom of group 3 material (such as boron, aluminum, etc.) has only 
three electrons which are available to take part in covalent bonds. Thus for 
every group 3 atom which is introduced into the crystal, there is one bond 
which is left unsatisfied and this introduces an unfilled energy state into the 
crystal. The position of this state on the energy band diagram is given in 
Fig. 1-19. When the group 3 atom acquires an additional electron from a 
nearby atom to satisfy all its covalent bonds it becomes a negative ion. The 
energy required for this process is small, and so the new energy state is just 
above the energy of the electron in its previous state, that is, just above the 



34 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



valence band. By an extension of the previous application of the Bohr theory 
of the atom this energy may be calculated. Table 1-3 shows the measured 
values of the energy level for various group 3 atoms present in germanium 
and silicon crystals (see "acceptor" in the table). 

At room temperature, the crystal has sufficient energy from thermal 
agitation to allow the energy state to be filled by an electron from the valence 
band. Figure 1-20 illustrates the condition that exists at room temperature. 
An electron from the valence band (i.e., an electron originally in a covalent 
bond somewhere in the crystal) gains sufficient energy to move into the state 
associated with the group 3 atoms. This leaves a positive hole in the valence 
band which is available for electrical conduction. The negatively ionized 



Conduction band 




Negatively ionized acceptor 
state (immobile) 



Positive hole free to move 
in valence band 



Distance — ► 

FIG. 1-20. The production of a positive hole in the valence band 
band of /Mype material. 

filled electron state is fixed in the crystal lattice and makes no contribution to 
the conduction process. The energy state associated with the group 3 material 
is called an acceptor state since it accepts electrons from the valence band. 

The addition of atoms of group 3 material causes a semiconductor to 
become p-type. In this case, positive holes are the majority carriers, and 
electrons are the minority carriers. The addition of only a small fraction of 
impurity atoms increases the conductivity by a large factor (as was the case for 
the f?-type material), but now the conductivity is mainly the result of the 
motion of positive holes in the semiconductor. 

We can now distinguish three types of semiconducting material. An 
intrinsic semiconductor is the pure group 4 crystal where the number of holes 
equals the number of electrons. Material is known as n-type when some atoms 
of group 5 are present in the crystal and there are more electrons than holes. 
When group 3 atoms are present, the material is p-type and holes predominate 
over electrons. 

Other distinguishing features of the three types of material will be pointed 
out in Chaps. 2, 3, and 4. In particular, we will show that a crystal containing 



SEC. 1-19 p- AND n-TYPE SEMICONDUCTORS 35 

a/7-type to /7-type junction (a p-n junction) has the electrical characteristics of 
a rectifier. 

The process of adding either donor or acceptor material to an otherwise 
pure crystal is referred to as doping the crystal. In preparation, the group 4 
material is purified to better than a few parts in 10 9 , and the required im- 
purity is added before the crystal is formed. If the same proportions of donor 
and acceptor material are added, the two effects cancel out, and the material 
becomes intrinsic again. It is the excess of one type of impurity over the 
other that determines the dominant type of conductivity (see Sec. 2-8). 

Figures 1-21 through 1-24 show the measured values of electron and hole 
mobilities in n- and/?-type germanium and silicon. The curves were obtained 
by measurement and calculation on semiconductor specimens of known 
resistivity. In each case, the mobility is plotted against the resistivity which is 
marked as an open scale at the top of the graph. The bottom scale is the 
density of impurity atoms in the material. N d signifies the number of donor 
atoms per cubic centimeter for n-type material, and N a is the number of 
acceptor atoms per cubic centimeter for /?-type material. These curves, 
therefore, allow us to determine the resistivity when the donor or acceptor 
density is known by comparing the scales at the top and the bottom of each 
graph. The equations relating mobility, resistivity, and impurity density are 
derived in Sec. 2-8. 

From Figs. 1-21 and 1-22, which relate to germanium, it can be seen that 
the addition of as few as 10 14 atoms of donor or acceptor material per cubic 
centimeter reduces the resistivity of the specimen appreciably. Since there are 
4.42 x 10 22 atoms of germanium per cubic centimeter, the impurity atoms 
form a minute proportion of the total number of atoms in the crystal but 
their effect on the resistivity is high. For both silicon and germanium, the 
curves show that the mobility decreases with decreasing resistivity for material 
of resistivity lower than about 10 ohm cm. 

1-20 Generation and Recombination 

Because of the vibrational energy of the crystal lattice at room temperatures, 
covalent bonds are continually being broken, and electrons are being set free. 
This process is known as the generation of electron-hole pairs. The reverse 
process is also taking place, and electrons are being recaptured by unfilled 
bonds to produce the recombination of electron-hole pairs. Both generation 
and recombination are random processes which are continuously occurring. 
When the temperature and external conditions are constant, a balance is set 
up between generation and recombination, and the resultant hole and electron 
densities are known as the thermal equilibrium values, p and n respectively. 
Since there are usually more than 10 10 carriers per cubic centimeter in a 
typical piece of semiconducting material, the statistical variation in these 
quantities is small and can be neglected. The symbols, p and n, are retained 
for the carrier densities under general conditions. 



36 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



3500 



■** 2500 
'o 
> 

g 2000 

.5 



10 13 



p in ohm cm 
1.6 



0.19 






10 1 



10 15 
N^ in cm 



10 1 



10 1 



FIG. 1-21. Electron and hole mobilities in «-type 
germanium at 300° K [from M. B. Prince, Phys. 
Rev. 92 (1953), 681 ; 93 (1954), 1204]. 



4000 
3500 
3000 
2500 



1500 



p in ohm cm 

3.5 0.40 0.07 



2 5 I 2 5 I 2 5 I 2 5 ' 2 5 



10 1 



10' 



10 1 " 10 1 

N„ in cm -3 



10' 



10' 



FIG. 1-22. Electron and hole mobilities in p-type 
germanium at 300° K [from M. B. Prince, Phys. 
Rev., 92 (1953), 681 ; 93 (1954), 1204]. 



1500 
1400 
1300 
1200 
1100 
a> 1000 

75 900 

| 800 

c 

* 700 

600 

500 

400 

300 

200 


20 


10 5 




2 

i 






1 




0.5 
















n-t 


ype 


silicon 








































































M„ 




















































































































_Mp_ 






















































7 I 


9 


2 


I 


-i 


E 


f 


> 7 { 


9 


2 



10 1 



10 J 



r d in cm 3 






1500 




1400 




1300 




1200 




1100 


o 


1000 


-< 




1 


900 




800 


_c 




a. 


700 



20 



FIG. 1-24. Electron and hole 
mobilities in />-type silicon at 
300° K [from M. B. Prince, 
Phys. Rev., 92 (1953), 681 ; 93 
(1954), 1204]. 



FIG. 1-23. Electron and hole 
mobilities in «-type silicon at 
300° K [from M. B. Prince, 
Phys. Rev., 92 (1953), 681; 
93 (1954), 1204]. 



p in ohm cm 

5 

















P- 


type 


j silicon 




















































































































J*n 










































































J^ 




















































6 7 I 


9 


* 


j 


A 




i 


7 i 


9 


2 



10 1 



10 16 



N„ in cm 



38 



ELECTRICAL CONDUCTION IN SEMICONDUCTORS 



CHAP. I 



It has been shown that the production of an electron-hole pair involves 
the passage of an electron from the valence band to the conduction band. 
Similarly, recombination is the process whereby an electron moves from the 
conduction band to the valence band. The direct transition of an electron 
from the conduction band to the valence band is unlikely, however, since a 
free electron and a hole will only recombine if momentum is conserved in the 
system, that is, if the electron and the hole initially have nearly equal and 
opposite momenta. The probability for this in germanium and silicon is low, 
and another recombination mechanism must be invoked. 



Conduction 
band 



-Trap 



Valence 
band 



Conduction 
band 



^Trap 



Valence 
band 



Distance 
(a) 



Distance 
(b) 



FIG. 1-25. (a) An electron moves from the conduction band into 
a recombination trap, (b) The electron moves from the trap down 
into the valence band. 

The majority of recombination events are found to take place at the site 
of an impurity atom or a lattice defect in the crystal. Such a location acts as 
a trap or recombination center and is, in effect, a third body which can satisfy 
the momentum requirements in the electron-hole collision. Figure 1-25 shows 
a trap which is located approximately in the center of the forbidden gap. 
The electron moves from the conduction band into the trap level and remains 
there for a short time before passing down into the valence band. Once an 
electron is in the trap it is not free to move and does not contribute to the 
conductivity. 

The average time that an electron spends in the conduction band before 
recombining via a trap is known as the lifetime, r n . Similarly, r p is the average 
lifetime of holes in the valence band. Electron and hole lifetimes are in the 
range 10" 8 to 10" 3 second for the semiconducting material used in transistors. 
The higher the conductivity the lower the lifetime since the added group 3 
or 5 atoms act as recombination centers. Lifetimes also depend upon the 
purity of the material (other than the presence of group 3 and 5 atoms), 
temperature, and shape and surface conditions of the specimen (see Sec. 9-10). 
In this book, we shall be primarily concerned with minority carrier lifetimes, 
for example, the lifetime of electrons in /?-type material. 



SEC. 1-20 



GENERATION AND RECOMBINATION 



39 



Consider unit volume of a piece of p-type material which has a thermal 
equilibrium electron and positive hole density n and p respectively. If 
excess electrons are introduced into the crystal, an equation can be derived to 
determine the way in which the density decays back to its original value. 
There is a generation rate, g, of electron-hole pairs per unit volume that is 
dependent upon the temperature for a given material but not upon the 
number of electrons present. If the electron lifetime is r n , and n electrons 
are present at time /, then the number of electrons recombining per second is 
n/r n . Thus, 



dn 



(1-30) 



where dn/dt is the rate of change of density with time (a negative quantity in 
this case). At equilibrium, dn/dt = and n = n , and so 



g = 



n 



(1-31) 



t=o 



n = Anexp(-t/r n ) 




Thermal 
equilibrium 
density n 



Time 



FIG. 1-26. The decay of excess carriers in a semiconductor 
[Eq. (1-33)]. 



Equation (1-30) can now be rewritten as 



dn 
It 



(1-32) 



and the solution is 



n = An exp I 1 



(1-33) 



Here, An is the value of n — n at t = 0, i.e., it is the excess electron density 
introduced into the material. Equation (1-33) and Fig. 1-26 show that the 
number of excess electrons in the conduction band decays exponentially with 
a time constant equal to the lifetime of the electrons in the material. 



40 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

By a similar argument, it may be shown that the corresponding equation 
for positive holes is 

p - Po = Ap exp (-—) (1-34) 

when excess positive holes, Ap, are introduced into «-type material, where r p 
is the lifetime of positive holes. 



BIBLIOGRAPHY 

General 

Dekker, Adrianus J., Electrical Engineering Materials, Englewood Cliffs, 
N.J.: Prentice-Hall, Inc., 1959 

Shive, J. N., Properties, Physics and Design of Semiconductor Devices, New 
York: D. Van Nostrand and Company, Inc., 1959 

Spangenburg, Karl R., Fundamentals of Electron Devices, New York: 
McGraw-Hill Book Company, Inc., 1957 

Sproull, Robert L., Modern Physics, New York: John Wiley & Sons, Inc., 
1956 

van der Ziel, Aldert, Solid State Physical Electronics, Englewood Cliffs, N.J. : 
Prentice-Hall, Inc., 1957 

Atomic physics 

Van Name, F. W., Jr., Modern Physics, 2d ed. Englewood Cliffs, N.J. : 
Prentice-Hall, Inc. 1962 

Wehr, M. Russell, and James A. Richards, Jr., Physics of the Atom, Reading, 
Mass.: Addison-Wesley Publishing Company, Inc., 1960 

Yarwood, John, Electricity, Magnetism and Atomic Physics, Vol. II, Atomic 
Physics, London: University Tutorial Press Ltd, 1958 

Crystals 

Dekker, Adrianus J., Solid State Physics, Englewood Cliffs, N.J.: Prentice- 
Hall, Inc., 1957 

Kittel, Charles, Introduction to Solid State Physics, 2d ed., New York: 
John Wiley & Sons, Inc., 1956 

Elementary semiconductor theory 

Dekker, Adrianus J., Electrical Engineering Materials, Englewood Cliffs, 
N.J.: Prentice-Hall, Inc., 1959 



BIBLIOGRAPHY 41 

DeWitt, David, and Arthur L. Rossoff, Transistor Electronics, New York: 
McGraw-Hill Book Company, Inc., 1957 

PROBLEMS 

l-l Write the expressions for the magnitudes of the tangential and angular 
electron orbital velocities as a function of the principal quantum 
number in the Bohr theory of the hydrogen atom. Calculate these 
velocities for n = 1. 

ans. 2.2 x 10 6 m sec -1 4.15 x 10 16 radsec _1 

1-2 Find the energy of an electron in the first, second, and third orbits of 
the Bohr atom in joules and electron volts. What are the wavelengths 
of the photons emitted when electrons make the transition between 
orbits (a) 1 and 2, (b) 2 and 3, (c) 1 and 3? 

1-3 Discuss the fundamental postulates used by Bohr in his theory of the 
hydrogen atom. Explain the meaning of the terms "excitation" and 
"ionization". 

1-4 Calculate the energy of one quantum of the following radiation: (a) 
X band radar (9300 Mc/sec), (b) far infrared, (c) blue light, (d) 50 
kilovolt X rays. 

ans. 3.8 x 10' 5 ev, 3 x 10~ 2 ev, 2.6 ev, 5 x 10 4 ev 

1-5 (a) Electromagnetic radiation of wavelength 949.5 A is projected on to 
hydrogen gas. Is any of the radiation absorbed? If absorption takes 
place, what are the initial and final energy states of the hydrogen atom? 
(b) Repeat the question for radiation of wavelength 2108.3 A. 

1-6 Before Bohr's theory of the atom was developed, Balmer gave an 
empirical relationship between the wavelengths of some of the spectral 
lines of hydrogen. This may be put in the form 



A = R \4 ~ ?) 



A 

where A is the wavelength of a spectral line in the series, n has the value 
3, 4, 5, 6, . . ., and R is known as Rydberg's constant. From Bohr's 
theory of the atom, explain which transitions produce this series of 
spectral lines and derive a value for R in terms of the fundamental 
constants. 

ans. R = (tne*/8e 2 h 3 c) 

Bohr's theory of the atom applies to any one-electron atom and can, 
therefore, be applied to singly ionized helium and doubly ionized 
lithium. What energy is required to remove the remaining electron 
from these two atoms, respectively? 



42 ELECTRICAL CONDUCTION IN SEMICONDUCTORS CHAP. I 

1-8 What is the de Broglie wavelength in Angstrom units of (a) an electron 
of energy 10 ev, (b) a proton of energy 1 million-electron volts, (c) a 
satellite of mass 100 lb orbiting the earth with a velocity of 18,000 mph? 

ans. 3.9 A, 2.86 x 10~ 4 A, 1.82 x 10" 29 A 

1-9 Explain why the discrete energy levels for a free, single atom will no 
longer apply to the same atom in a crystal. Distinguish among con- 
ductors, insulators, and semiconductors. 

I — 10 If the lattice constant for Fig. 1-8 is 5.66 A, what is the distance be- 
tween neighboring atoms? 

1-1 1 A germanium crystal specimen 1 mm cube has a total of 2.5 x 10 7 
electrons in its conduction band. What electron current flows when 
there is a field of 6 volts cm -1 parallel to one face of the cube? 

1-12 One atom of boron is contained inside an otherwise pure germanium 
crystal. Using the concepts of covalent binding between neighboring 
atoms in a crystal, in your own words explain the effect of the added 
boron atom on the electrical properties of the crystal. 



2 



Electrons The previous chapter presented a descriptive 
account of electrical conduction in semiconductors. 
This chapter provides a more exact analysis of the 
behavior of electrons and holes. This is carried out 
by using an elementary treatment of Fermi-Dirac 
statistics. 



and Holes in 
Semiconductors 



44 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

2-1 The Density of States in an Energy Band 

The energy band representation which has been used in Chap. 1 shows that 
there are electron energy states in both the valence and conduction bands in 
a semiconductor and that these bands are separated by a forbidden gap. In 
the valence band of a group 4 semiconductor, it is known that there are four 
electron states for each atom (Fig. 1-10), and so the total number of states is 
four times the number of atoms. This makes it possible to establish the total 
number of states in the valence band per unit volume but gives no information 
on the number of holes and electrons in the material. 



Momentum in 
y direction 




Momentum in 
z direction 



\ Momentum in 
I x direction 



(a) 



(b) 



FIG. 2-1. A momentum cell diagram, (a) Cube of metal having 
sides Ax = Ay = Az = 1. (b) Momentum diagram showing unit 
cell of dimensions Ap x = Ap y = Ap z = h. Dotted circle indicates 
sphere of radius p M inside which all cells are occupied when M 
electrons per unit volume are in the band at 0°K. 



An electron-hole pair is produced when an electron acquires sufficient 
energy to leave the valence band and enter the conduction band. The passage 
of an electron from the top of the valence band to the bottom of the con- 
duction band requires the expenditure of the least amount of energy of any 
transition which can be made between the two bands. Transitions will 
normally be of this type, and so the number of electrons which move between 
the two bands depends partly on the temperature and partly on the number 
of electron energy states at the two band edges. It is, therefore, important to 
learn how the states are distributed with respect to energy in the valence and 



SEC. 2-1 THE DENSITY OF STATES IN AN ENERGY BAND 45 

conduction bands. We will first consider the simple case of the energy states 
in the outer (conduction) band of a metal. In Sec. 2-2, this result will be 
applied to the valence and conduction bands in a semiconductor. 

Consider a cube of metal having unit sides in the three coordinate direc- 
tions as shown in Fig. 2-l(a). From Sec. 1-15, the electrons in the outer 
energy band of the metal are free to move within the limits of the unit cube. 
Thus, an electron cannot be located more accurately than to state that it is 
still in the cube, that is, its position is specified to within Ax, Ay, and Az in 
Fig. 2-1 (a), where 

Ax = Ay = Az = I (2-1) 

The momentum of the electron will next be considered. If p x , p y , p z are the 
three components of the momentum in the three coordinate directions, we 
are able to specify the momentum to within Ap x , Ap y , and Ap z by applying 
Heisenberg's uncertainty principle as expressed in Eq. (1-16). Thus, 

Ap x = Ap y = Ap 2 = h (2-2) 

This application of the uncertainty principle is illustrated in Fig. 2-1 (b) 
where the three axes indicate momentum in the three coordinate directions, 
and the vector, p, represents the magnitude and direction of the momentum 
of any one electron. The uncertainty principle shows that, for a unit cube of 
the metal, it is impossible to specify p with any more accuracy than to state 
that the line representing p must end somewhere within a cell of side h and 
volume h 3 on the momentum diagram. This cell is known as the unit mo- 
mentum cell, and it can be seen that other cells having the same dimensions 
but different positions can be drawn to represent the accuracy of specification 
of the momenta of other electrons in the metal. 

This pictorial representation of momentum cells can be extended by 
applying the Pauli exclusion principle. In Sec. 1-10, it was stated that no 
two electrons can have the same set of four quantum numbers in an atom 
and, because of the close coupling between atoms, the exclusion principle 
applies in the metal crystal as well. Since the spin quantum number s can 
take only the values + \ or —\, the exclusion principle may be restated in 
the form that only two electrons can have the same three quantum numbers 
n, I, and m. The principle is now applicable to the momentum diagram since 
the three quantum numbers determine the momentum p. Thus each unit 
momentum cell, which indicates the accuracy of specifying one value of p, 
now accounts for two electrons having opposite spins (s — + \ and s = —J). 

Consider now that there are M free electrons available in the material. 
At a temperature of 0° K, there is no disturbing motion in the crystal due to 
heat energy, and the electrons will occupy cells having the lowest values of 
momentum (and energy). There is no preferred direction in the crystal, and 
so M/2 cells arranged spherically about the origin as center up to a radius p M 
will be occupied as shown in Fig. 2-1 (b). The "volume" of each cell is h 3 



46 



ELECTRONS AND HOLES IN SEMICONDUCTORS 



CHAP. 2 



and so the total "volume" of occupied cells in the momentum diagram is 



h 3 M 



= 3^m 



(2-3) 



The term "volume" has been used, and we note that the axes of Fig. 2-l(b) 
have the dimensions of momentum not length. Figure 2-1 (b) is often called 
a momentum space diagram. 

It is possible to relate the maximum momentum, p M , with the position of 
the corresponding electron energy state, E. On the momentum space dia- 
gram, the zero of momentum (p = 0) corresponds to the bottom of the 



Area represents 

number of states 

dM in energy 

range dE 




Energy density of states S(E) 



FIG. 2-2. Energy density of states plotted as a function of energy. 



energy band which we will call E c . The energy difference between the levels 
at E and E c corresponds to the momentum p M , or 



1 2 Pm 
2«* = 2^ 



(2-4) 



where m is the mass of the electron, and v M is its velocity. Eliminating p M 
between Eqs. (2-3) and (2-4), 



M = 



,/2 m 3/2 77- 

3/2 3 



(e - E c y 



(2-5) 



Equation (2-5) gives the relationship between the total number of electron 
states which are filled at 0° K and the energy level of the highest state relative 
to the bottom of the band, E - E c . If M is increased by a small value, dM, 
the radius of the sphere on the momentum space diagram will increase from 
Pm to/? M + dp M , and the highest filled energy level will rise from £to E + dE. 



SEC. 2-1 THE DENSITY OF STATES IN AN ENERGY BAND 47 

We now define the energy density of states to be S(E) given by the equation 

S(E) = 4£ (2-6) 

where the E in parenthesis indicates that the energy density, S, is a function 
of energy. The energy density is thus the number of states, dM, present in 
unit energy range, and indicates the degree of '-packing" of the states with 
respect to energy. This is the most useful way of expressing the information 
given in Eq. (2-5) since it gives the numbers of states per unit energy at any 
level in the band. By differentiation, 

dM 2 7/2 m 3/2 7r 
S(E) = ~^= ™ 3 (E ~ E c ) 1!2 per unit volume (2-7) 

Equation (2-7) has been plotted in Fig. 2-2. Energy has been taken as the 
ordinate in this curve to conform with the energy band diagrams given 
previously. The shaded area in the figure is equal to the number of states, 
dM, in energy range dE in accordance with Eq. (2-7). This information, 
which we have derived for the simple case of a metal, will now be applied to 
the conduction and valence band in a semiconductor. 



2-2 Energy State Densities 

in Semiconductors and the Effect of Temperature 

The energy density of states at the bottom of the conduction band of a semi- 
conductor is given by 

S(E) = ™ c (E - E c y 2 (2-8) 

This is of similar form to Eq. (2-7) with E c now the energy at the bottom of 
the conduction band and m c as the effective mass of the electron in that band. 
It is necessary to introduce the concept of effective mass to take into account 
the influences of the neighboring atoms and charges on the motion of an 
electron in the band. Thus, an electron will respond to an applied force as if 
it had a mass of m c (not the free electronic mass, m) when it is in the conduc- 
tion band. In the valence band, the effective mass of the positive hole is written 
as m v , and by analogy with the argument of the previous section, we write the 
energy density of states in the valence band as 

97/2 m 3/2 

S(E) = Z ™ v V (E V -Ey>* (2-9) 

where E v is the energy at the top of the valence band, and E is any energy in 
the valence band. The values of m c and m v are not the same but they will be 
considered so in this simplified analysis. Note that this introduces a small 



48 



ELECTRONS AND HOLES IN SEMICONDUCTORS 



CHAP. 2 



Conduction band 



error in Eq. (2-22) (see Problem 2-7). The curve of S(E) versus is is given in 
Fig. 2-3 for the valence and conduction bands of a semiconductor. 

The production of hole-electron pairs was shown in Chap. 1 to be the 
result of the passage of an electron from the valence band to the conduction 
band. In this chapter, we look at the same phenomenon from a different point 
of view and consider how many of the electron energy states in the two bands 
will be occupied. To do this, we introduce the Fermi factor, F(E), which is a 
number that expresses the probability that a state of a given energy is occupied 

by an electron. This number has a value 
between zero and unity, and is a func- 
tion of energy and temperature. Zero 
probability means that the state is 
empty, and a probability of unity implies 
that the state is occupied. A probability 
of 0.5, for instance, means that states at 
this energy level are 50 per cent occu- 
pied. Since this is a statistical process, 
this probability represents only the 
average occupancy rate. However, the 
total number of states involved is usually 
so high that the fluctuation from the 
statistical average is small. 

The Fermi factor, F(E), can be 
derived by a statistical argument which 
will not be given here (see the biblio- 
graphy at the end of this chapter). In 
essence, it consists of finding the most 
probable arrangement of electrons in 
energy states, when occupancy is governed by the Pauli exclusion principle, 
when there is a given total energy in the system and when the number of 
electrons is a constant. Under these circumstances the Fermi factor is 



Forbidden gap 



Valence band 



S(E) -+ 

FIG. 2-3. Energy density of states in 
valence and conduction band. 



F(E) = 



1 + exp [(E - E F )/kT] 



(2-10) 



where E F is a particular energy known as the Fermi level, k is Boltzmann's 
constant [1.38 x 10" 23 joule CK)" 1 ], and J is the absolute temperature, °K. 
The Fermi factor is independent of the energy density of states; it is the 
probability that states are occupied at that level irrespective of the number of 
states actually present, that is, it is the fractional occupancy of possible states. 
Even if there are no states at a particular energy (as in the forbidden gap of an 
intrinsic semiconductor), the Fermi factor still has a definite value. 
Before discussing the Fermi factor further, consider Eq. (2-11). 



N(E) dE = S(E)F(E) dE 



(2-11) 



SEC. 2-2 



DENSITIES IN SEMICONDUCTORS AND THE EFFECT OF TEMPERATURE 



49 



Since S(E) is the energy density of states, S(E) dE is the total number of 
available states in the energy range dE. E{E) is the probability that these states 
are occupied and so the product of the density of available states, S(E), and the 
probability of their being occupied, E(E), gives the number of occupied states, 
N(E), in the small energy range considered, dE. Equation (2-11) and this 
verbal statement of it are important for an understanding of later work in this 
chapter. 




^-T=o°K 



Fermi factor F(E) — * 
(probability that states are occupied) 

FIG. 2-4. Fermi factor as a function of energy for two different 
temperatures. 

The value of the Fermi factor is determined by the ratio (E — E F )/kTfrom 
Eq. (2-10). The quantity kT has the dimensions of energy and can be specified 
in electron volts. For example, kT = 25.8 millielectron volts at 300° K. The 
variation of E(E) with energy is shown in Fig. 2-4. As the temperature is 
increased, the size of the units of energy shown in the figure also increases 
showing that F(E) is a function of temperature. The Fermi level, E F , is a 
quantity of great significance in semiconductor electronics. It is a reference level 
which is an important characteristic of a semiconductor as the following 
analysis will show. 

When the temperature is 0° K, F(E) can have one of two values. 



For E > E F , 

For E < E F , 



F(E) = 
F(E) = 



1 + exp ( oo) 
1 



1 + exp ( — oo) 



= 
= 1 



50 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

Therefore, the value of F(E) is unity for all energies below the Fermi level and 
zero for all energies above the Fermi level. This corresponds with the case 
shown in Fig. 2-1 where all the momentum cells were full up to the radius 
p M at 0° K. The energy corresponding to p M must coincide with the Fermi 
level, and we can say that at 0° K all energy states will be full up to E F . 

A more useful definition of the Fermi level can be obtained by putting 
E — E F at any temperature other than 0° K. Then, if E = E F and T ^ 



1 + exp (0) 2 

Thus, the Fermi level, E F , is that energy where the probability of a state being 
occupied is \. 

Referring to Fig. 2-4, at 0° K all the energy states below the Fermi level 
are completely occupied whereas above this level they are completely empty. 
At 0° K there is no heat energy present in the crystal lattice, and so no cova- 
lent bonds are being broken. At any other temperature, however, the process 
of generation and recombination of electron-hole pairs is taking place. 
Because of this, there is a probability of slightly less than one that all of the 
states below the Fermi level are occupied, and also a small probability that 
some of the energy states just above the Fermi level are occupied. 

It has been shown that both electrons and holes contribute to the con- 
ductivity of semiconductors. In the conduction band, therefore, the prob- 
ability of electron states being occupied, F(E), is of interest because all 
electrons in this band are mobile. In the valence band, positive holes are 
responsible for the passage of charge. Thus, it is the absence of electrons 
which is important in the valence band, and we shall be considering 1 — F(E) 
since this is the probability of a positive hole, rather than a filled state, 
occurring at a particular energy level in the band. 

The concept of the probability of occupancy of an energy state, as 
represented by F(E) is applicable to all solid-state crystalline materials. 
Equation (2-10) is true for conductors, insulators, and semiconductors. 
The position of the Fermi level depends upon the available number of elec- 
trons in the material and upon the energy state density curve, S(E). Relative 
to the Fermi level, however, the probability of an energy state being filled 
depends only upon the temperature as shown by Eq. (2-10). 

The total number of electrons per unit volume in a material is M where 

M = JN(E) dE = JS(E)F(E) dE (2-12) 

and the limits of integration are from the bottom of the lowest energy band 
to the top of the highest band concerned. Equation (2-12) can be used to 
find the position of the Fermi level for any material when the total number of 
electrons per unit volume, M, is known and where S(E) can be determined. 
The value of the Fermi level gives important information about a solid 



SEC. 2-2 



DENSITIES IN SEMICONDUCTORS AND THE EFFECT OF TEMPERATURE 



51 



material since it indicates the highest possible filled electron energy state at 
0° K. At any other temperature, as shown in Fig. 2-4 and Eq. (2-10), the 
value of F(E) drops from 0.95 to 0.05 as the energy changes from E F — 3kT 'to 
E F + 3kT. Below this region, states are almost completely full; above this 
region, they are almost empty. The Fermi level is in the center of this region. 
The ideas presented in this section will be applied to semiconductors to 
determine the number of electrons in the conduction band, the number of 
holes in the valence band, and the position of the Fermi level. The case of an 
intrinsic semiconductor will be considered first. 

2-3 Intrinsic Semiconductor 

In a specimen of unit volume, the total number of occupied states with 
energy between E and E + dE is equal to the product of the density of the 



t 


^/Conduction 








^y^ band 




^> 


bq 
& 


E F 














a 










w 


^v Valence 
band 


i 


(Expanc 


ed scale) 



S(E) — 
Energy state density 



0.5 

F(E) -* 
Fermi factor 



1.0 



— P(E) N(E) — * 

Density of Density of 

positive holes electrons in 

in valence conduction 

band band 



FIG. 2-5. Graphical illustration of the number of electrons and 
holes in an intrinsic semiconductor showing the position of the 
Fermi level. 



available number of states and the probability of their occupancy. Equation 
(2-11) is repeated here for convenience. 



N(E) dE = S(E)F(E) dE 



(2-11) 



This relationship is illustrated in Fig. 2-5. The left-hand curve is a plot of 
Eqs. (2-8) and (2-9) and shows the density of available electron states close 
to the forbidden gap for both the valence and conduction bands. The central 
curve shows the Fermi factor at room temperature, and the right-hand curve 
is obtained by taking the product of the values shown in the two preceding 
curves at all energy levels. The density of electrons in the conduction band as 
given by Eq. (2-11) is shown to the right of the energy axis. The density of 



52 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

positive holes in the valence band is shown to the left of the energy axis. 
This number can be represented by P(E) dE in an equation which uses 
1 - F(E) in place of F(E) in Eq. (2-11), i.e., 

P(E) dE = F(E)[l - F(E)] dE (2-13) 

Under certain conditions an approximation to the Fermi factor can be 
made. For a room temperature of 300° K, for example, the value of kT is 
25.8 millielectron volts. At an energy level E x which is more than 100 milli- 
electron volts (say) above the Fermi level, 

and Eq. (2-10) can be written as 

F(E0 ~ exp (^~i) (2-14) 

The right-hand side of Eq. (2-14) is known as the Boltzmann factor because it 
corresponds with the relationship obtained from classical Maxwell-Boltzmann 
statistics, which apply to a perfect gas. We see from Eq. (2-14) that the 
probability of occupancy drops exponentially with energy at levels well above 
E F . 

The probability of finding an electron state unoccupied in the valence 
band has already been given as the difference between unity and the Fermi 
factor at that energy. Assuming E 2 is an energy level in the valence band 
which is far removed from the Fermi level, 

1 - w = » - 1 + exp id - mm (2 " 15) 

where E F - E 2 » kT. In Eq. (2-15), the exponential term is much less than 
unity, and so by the binomial expansion, 

1 - F(E 2 ) x 1 - [l - exp (^f^)] 

= exp {^^) (2-16) 

Equation (2-16) indicates that as E 2 becomes more negative the probability 
of finding holes decreases exponentially. 

This information allows us to calculate the position of the Fermi level in 
an intrinsic semiconductor. First, it should be noted that the Fermi level 
will be somewhere in the forbidden gap since nearly all the valence band 
states are occupied, and only very few electrons will be found in the conduc- 
tion band at room temperatures. Next, Eqs. (2-8) and (2-9) give the required 
information of the energy state densities in the two bands, and finally, Eqs. 



SEC. 2-3 



INTRINSIC SEMICONDUCTOR 



53 



(2-14) and (2-16) give the approximate probability factors to be used in the 
calculation. 

Consider energy levels E 1 in the conduction band and E 2 in the valence 
band symmetrically placed about the center of the gap. For an energy range 
dE in both cases, the number of electrons in the conduction band is 

NiEJ dE = S(E 1 )F(E 1 ) dE (2-17) 

and the number of positive holes in energy range dE in the valence band is 



P(E 2 ) dE = S(E 2 )[l - F(E 2 )] dE 

(2-18) 



Assuming for simplicity that m c 
in Eqs. (2-8) and (2-9), 



Ey 



m, 



Thus, 



S(E ± ) = S(E 2 ) 

N(EJ _ FjEJ 
P(E 2 ) 



(2-19) 



F(E 2 ) 





_L 


y//////// 


dE 


X f 

y"^ Conduction band 
Center of gap 


>v Valence band 


V//////7S. 


dE 


\ | 



_ exp [(E F - EJ/kT] 
exp [(E 2 - E F )/kT] 
(2-20) 

assuming E x — E F » kT and E F - E 2 
» kT. 

Equation (2-20) applies to all sym- 
metrically placed elements on Fig. 2-6. 
Hence, the ratio of the numbers of elec- 
trons and positive holes for any element 
is the same as the ratio of the total 
numbers of electrons and positive holes 
in the semiconductor, which is unity for intrinsic material. Thus, 



E< 



S(E) -+ 

FIG. 2-6. Curve of S(E) versus E for 
an intrinsic semiconductor showing the 
location of elements spaced symmetric- 
ally about the center of the forbidden 
gap. 



Therefore, in Eq. (2-20) 



P(E 2 ) Pt 



Hip Hii Hi 2 



(2-21) 



kT 



or 



E P = 



kT 
E 2 + E 1 



(2-22) 



It will be recalled that the energy levels E 1 and E 2 were taken symmetrically 
about the center of the forbidden gap. Equation (2-22) shows that the Fermi 
level lies at the center of the forbidden gap for intrinsic material and that it is 
independent of temperature. The error introduced by putting m c = m v is 
small in practice (see Problem 2-7). 



54 



ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 



2-4 n-type Semiconductor 

When atoms of group 5 material are introduced into an otherwise pure 
crystal of germanium or silicon, additional energy levels are introduced just 
below the bottom of the conduction band in the forbidden gap. The curve of 
available energy states versus energy is no longer symmetrical about the 
center, and the Fermi level is no longer at the center of the gap. 



! 


Conduction 
band 










^^y^ Donor 
r*-^ states 

~Jae~ ** 


— -■— — 


< 


^ 




Valence 
band 




(Expanded scale) 






0.5 1.0 


I 


S(E) -+ 
Energy state density 


F(E) -* 
Fermi factor 


*- P(E) 

Density of 

positive holes 

in valence 

band 


N(E) -+ 
Density of 
electrons in 
conduction 

band 



FIG. 2-7. The position of the Fermi level in an n-type semi- 
conductor. 



In Fig. 2-7, the Fermi level is shown at an energy AE above the center of 
the gap. By comparison with the case of the intrinsic semiconductor where 
the Fermi level is located at the center of the gap, the Fermi factor is increased 
by the ratio exp (AE/kT) everywhere in the conduction band [see Eq. (2-14)]. 
Thus, the total number of electrons in the band (which was n { for the intrinsic 
case) is now given by 



n = n i exp 



m 



(2-23) 



For an «-type semiconductor, n > n h and so AE is positive, i.e., the Fermi 
level for an «-type semiconductor lies above the center of the gap. 

The energy level of the donor states lies just below the bottom of the 
conduction band, and the Fermi factor at this energy is low. The donor states 
will, therefore, be almost empty, and the donor electrons will be found in the 
conduction band. Note that the probability of occupancy in the conduction 
band is even lower than in the donor states, but there are many more states 
present in the conduction band. 



SEC. 2-4 



n-TYPE SEMICONDUCTOR 



55 



When the density of donor atoms in the material is high, the conductivity 
is almost entirely due to donor electrons in the conduction band as explained 
in Sees. 1-19 and 2-8. Under these conditions, 



N d = n i exp 



m 



(2-24) 



are 



and so the position of the Fermi level can be calculated if N d and n { 
known. 

The number of holes in the valence band is small in /7-type material. This 
is shown diagrammatically in Fig. 2-7 where it is seen that raising the Fermi 
level above the center of the forbidden gap decreases the value of 1 — F(E) 
at the top of the valence band. 



2-5 p-type Semiconductor 

When group 3 atoms are introduced into a germanium or silicon crystal, the 
acceptor energy level is just above the valence band in the forbidden gap. 
The number of positive holes in the valence band is greater than the number of 
electrons in the conduction band, and the Fermi level is below the center of 
the forbidden gap as shown in Fig. 2-8. 




w 



JA& 

>"*- Acceptor 
states 




1.0 



S(E) -* 
Energy state density 



F(E) - 
Fermi factor 



(Expanded scale) 



<— P(E) N(E) — 

Density of Density of 

positive holes electrons in 

in valence conduction 

band band 



FIG. 2-8. The position of the Fermi level in a p-type semi- 
conductor. 



If AE' is the energy by which the Fermi level is displaced down from the 
center of the gap in this material, the total number of holes in the valence 
band can be written as 

p = n t exp \j^j (2-25) 

by comparison with the arguments of the previous section. For material 



56 



ELECTRONS AND HOLES IN SEMICONDUCTORS 



CHAP. 2 



which has so many added acceptor atoms that the conductivity is mainly due 
to holes, p = N a , and 



N a 



n { exp 



(w) 



(2-26) 



The number of electrons in the conduction band is small, as shown in Fig. 2-8. 

2-6 Variation of the Fermi level with Temperature 

For intrinsic material, raising the temperature increases the number of both 
electrons and positive holes since more energy is available to allow a valence 
electron to break its bond. The number of electrons is still equal to the num- 
ber of positive holes, however, and so the Fermi level remains approximately 
at the center of the forbidden energy gap whatever the temperature. 



Intrinsic 



n-type 



p-type 



Conduction band 



-Fermi level 
Fe_rmiievel_,/j unchange d 

with 
temperature 



+ ♦ ♦ + 

Valence band 



Conduction band 



o o o o o o - 



Valence band 



Donor 
states 
.T=20°C 



at 



N T=100°C 

Acceptor >* 
st.at.ps y 



Conduction band 



T=20°C 



,o o o o 



♦ + + + ♦ ♦ 

Valence band 



Distance 



Distance 



Distance 



FIG. 2-9. Effect of temperature on Fermi level for typical in- 
trinsic n- and /?-type semiconductors. 

In an «-type crystal, there are electrons in the conduction band that have 
come from two different sources. Some of the electrons will be those from the 
group 5 impurity atoms, and these are easily separated from their parent 
atom. Their number does not vary much as the temperature is altered within 
the range of temperatures where semiconducting devices are commonly used 
(0° to 50° C). The other electrons in the conduction band are present because 
of the breaking of a covalent bond. Such "intrinsic effect " electrons will 
increase in number as the temperature is raised, and so their proportion of the 
total number will increase. Thus as the temperature rises, the material 
becomes more intrinsic and the Fermi level moves closer to the intrinsic 
position, that is, to the center of the forbidden gap. Hence as the temperature 
is raised, the material becomes increasingly nearer to intrinsic material in its 
electrical properties. 

For p-type material, the Fermi level at room temperature is below the 
center of the forbidden gap. As the temperature is increased, this material 



SEC. 2-6 VARIATION OF THE FERMI LEVEL WITH TEMPERATURE 57 

also becomes increasingly intrinsic for the same reason, and the Fermi level 
rises until it approaches the center of the gap. Thus both p- and tf-type 
material become more like intrinsic material at high temperatures. This 
places a limit on the operating temperature of a semiconducting device. 



2-7 The Law of Mass Action 

The number of electrons in an energy range dE in the conduction band at a 
level E has previously been given as 

N(E) dE = S(E) exp {^f^) dE (2-27) 

where E — E F » kT. The total number of electrons in the conduction band 
is 

n= f N(E) dE = J°° S(E) exp ( Ep ~ T E \ dE (2-28) 

The upper limit of the integral is put as infinity for convenience in the 
integration. This limit will certainly include all the electrons in the conduction 
band. Now the value of S(E) has been given previously as 

S(E) = K C (E - E c ) 112 (2-8) 

for the bottom of the conduction band only, where 

K c = 2 7/2 m 3/2 77/r 3 

The exponential term of Eq. (2-28) dominates the equation for energies more 
than a few kT above the bottom of the conduction band, and the value of the 
integrand soon drops to zero upon moving upwards through the band. 
Hence, all the electrons will be found at the bottom of the band, and so the 
limit can be used even when Eq. (2-8) is substituted in the integral. Thus, 

n = K c £ (E - E c y* exp (^r^) dE 

= K c exp [^fi) f™ (E - E C Y* exp (^^) dE (2-29) 
This integral may be put in the form 



r 00 77- 1/2 

x ll2 e- x dx = ^- 
J ^ 



and so 



n = l^K^k^T* 12 exp ( ^Jf^ ) (2-30) 



58 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

Similarly for holes in the valence band it can be shown that 

p = 2-%7r 1 ' 2 A: 3/2 r 3/2 exp ( Ev ~ T Ep \ (2-31) 

where 

K v = 2 7/2 m 3/2 7r/r 3 

Multiplying Eq. (2-30) by Eq. (2-31), 

= ^r 3 exp(^) (2-32) 

where E g = E c — E Vi and is the width of the forbidden gap, and 

K 2 = AcAp — (2-33) 

From Eq. (2-32), the product np is seen to be independent of the position of 
the Fermi level and is thus the same value for p-type, n-type or intrinsic material. 
Using the subscript i for intrinsic material, 

np = n { pi = n 2 (2-34) 

Equation (2-34) is known as the law of mass action. From Eqs. (2-32) and 
(2-34), 

» I -Jf i r»«exp(^) (2-35) 

which allows us to calculate n { for a given material if K x and E g are known. 
In the previous sections of this chapter, we put m c = m v without incurring 
serious error. In this calculation, however, the product of m c and m v is 
involved in the constant K ± of Eq. (2-35), and m c cannot be equated to m v . 
If the energy gap and the intrinsic density of the carriers are known by 
experiment, the product of the effective masses can be calculated from Eq. 
(2-35) (see Problem 2-8). 

For germanium at 300° K, E g = 0.72 ev, and n t = 2.5 x 10 13 cm" 3 . For 
silicon at 300° K, E g = 1.1 ev, and n t = 1.6 x 10 10 cm" 3 . 

2-8 Mobile and Immobile Charges 

Consider a piece of semiconducting material containing atoms of both group 
3 and group 5. Let 7V a and N d be the density of atoms of the two groups 
respectively. The material will be />-type or «-type according to which kind 
of impurity atom predominates. Let the density of positive holes in the 
material be p and the density of electrons be n. 



SEC. 2-8 MOBILE AND IMMOBILE CHARGES 59 

In germanium or silicon at room temperature it is found that the energy 
state of the fourth (missing) valence electron of the group 3 (acceptor) atom 
is sufficiently close to the valence band that every acceptor ion has an extra 
electron associated with it (see Sec. 1-19 and Fig. 1-20). Thus there is a 
resultant negative charge ( — e) associated with each acceptor atom in the 
crystal lattice at room temperature. Similarly, every group 5 donor atom 
donates its extra electron into the conduction band and a positive charge 
( + e) is associated with every donor atom (see Fig. 1-18). 

The crystal as a whole must be electrically neutral since it was formed from 
neutral group 3, 4, or 5 atoms. Therefore, a charge neutrality equation may 
be written summing the negative charges as ne + N a e and the positive charges 
as pe + N A e. Hence, 

n + N a =p + N d (2-36) 

From Eq. (2-34), 

P 

Hence, ^ + N a = p + N d 

p 

or p* + (N d - N a )p - n? = (2-37) 

The solution of Eqs. (2-34) and (2-36) is, therefore, 



~(N d - N a ) ± V(N d - N a ) 2 + 4/i? 



n = (N d - N a ) ± V(N d - N a f + 4/2? (2 _ 3g) 

Assume that the material is «-type and that A^ - N a » 2n t . Then in Eq. 
(2-38) the positive sign in front of the square root is selected since n must be 
positive. Hence, 

n « N d - N a 
and from Eq. (2-34), 

1 (2-39) 



N d - N, 



When the density of acceptor atoms is higher than donor atoms in the 
crystal, and assuming that N a - N d » 2n h Eq. (2-38) becomes 

p = N a - N d 
and n = ^- (2-40) 

Equations (2-39) and (2-40) show that it is the excess of one type of 
impurity density over the other that determines the type of material. When 



60 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

N a = N d , the material becomes intrinsic again provided the donor and accep- 
tor atoms are not present in such numbers as to destroy the lattice properties 
of the group 4 crystal. 

For n-typt material where no acceptor atoms are present and N d » 2n u 

n = N d 

P = i (2-41) 

Since N d is relatively high, the contribution of the minority carriers to the 
conductivity can be ignored, and 

a = ne\ji n + pefM p X N d e^ n (2-42) 

For strongly />-type material where no donor atoms are present, the 
corresponding equations are 

P = N a 



'I 

N„ 



n = %■ (2-43) 



and a = ne[x n + pe\i v X N a e[x p (2-44) 

Example : 

Germanium has a donor-type impurity added to the extent of one atom per 
10 8 germanium atoms. What effect does this have on the conductivity of the 
material at 300° K? 

For intrinsic germanium at 300° K from Table 1-2 and Figs. 1-21 and 
1-22, 

nf = 6.25 x 10 26 cm- 3 
/x n = 3900 cm 2 volt" 1 sec" 1 
Ijl p = 1900 cm 2 volt" 1 sec" 1 

The conductivity of intrinsic germanium is 

Oi = n^n + ii p ) 

= 2.5 x 10 13 (1.6 x 10" 19 )5800 

= 0.0232 ohm" 1 cm" 1 

There are 4.42 x 10 22 atoms of germanium per cubic centimeter (see Appen- 
dix), and so the number of donor atoms per cubic centimeter is 4.42 x 10 14 . 
N d is, therefore, about 18 times n h and we may use the simplified form of 
Eq. (2-38) without introducing an error of more than 1 per cent. This is not 
significant, since semiconductor properties can seldom be measured with an 



5EC. 2-8 MOBILE AND IMMOBILE CHARGES 

accuracy of better than 3 per cent. Thus, 

n = N d = 4.42 x 10 14 cm- 3 
6.25 x 10 26 



61 



P 



nf 



1.41 x 10 12 cm 



N d 4.42 x 10 14 
and Eq. (2-42) becomes 

o a N d efi n = 4.42 x 10 14 (1.6 x 1CT 19 )3800 
= 0.269 ohm" 1 cm" 1 
where the mobility is obtained from Fig. 1-21. 



This example illustrates several important points. First, the addition of 
only 1 part of donor impurity to 10 8 parts of germanium increases the 
conductivity of a crystal approximately ten-fold, showing how sensitive the 
conductivity is to donor (or acceptor) type impurity. Secondly, although 
this type of calculation aids us in understanding the mechanism of semi- 
conductors, it is difficult to carry out for high impurity densities where the 
mobility is a function of the conductivity. Finally, the experimental results 
given in Figs. 1-21 through 1-24 provide a ready-made solution to this type 
of problem. 

2-9 The Hall Effect 

It is often necessary to determine whether a material is /7-type or /?-type. 
Measurement of the conductivity of a specimen will not give this information 



Probe attached 
to face 2 




Direction of 
conventional 
current flow 



Probe attached 
to face 1 



FIG. 2-10. Illustration of the Hall effect. 



62 



ELECTRONS AND HOLES IN SEMICONDUCTORS 



CHAP. 2 



since it cannot distinguish between positive hole and electron conduction. The 
Hall effect can be utilized to distinguish between the two types of carrier, and it 
also allows the density of the charge carriers to be determined. In conjunction 
with a measurement of conductivity the mobility of the carriers can be found. 
In 1897, Hall discovered that when a current was passed through a slab 
of material in the presence of a transverse magnetic field, a small potential 
difference was established in a direction perpendicular to both the current 
flow and the magnetic field. Figure 2-10 shows the current flowing in the 
positive x direction with the flux density, B, in the positive z direction and the 
potential difference appearing in the y direction. Probes may be attached to 
faces 1 and 2 and the potential difference may be measured by a high sensitivity 
d-c vacuum tube voltmeter of very high input resistance. 



Face 2 




Face 1 
Direction of B out of paper 



FIG. 2-11 
effect). 



Motion of electrons in an //-type semiconductor (Hall 



The Hall effect may be explained by reference to Fig. 2-1 1 which shows 
the front face of the slab only. Assuming that the material is an «-type 
semiconductor, the current flow consists almost entirely of electrons moving 
from right to left. This corresponds to the direction of conventional current 
from left to right as in Fig. 2-10. 

If v is the velocity of electrons at right angles to the magnetic field, there 
is a downward force on each electron of magnitude Bev. This causes the 
electron current to be deflected in a downwards direction and causes a nega- 
tive charge to accumulate on the bottom face of the slab (face 1). A potential 
difference is therefore established from top to bottom of the specimen with 
the bottom face negative. This potential difference causes a field $ in the 
negative y direction, and so there is a force of e$ acting in the upward direc- 
tion on the electron. Equilibrium occurs when 



or 



= Bev 
= Bv 



(2-45) 



SEC. 2-9 THE HALL EFFECT 

The Hall coefficient is defined to be 



63 



*■" ~BJ 



(2-46) 



where J is the current density of the electron stream, and the negative sign is 
used because the electric field is in the negative y direction. Now J = nev, 
thus 



Rn = 



BJ 



v 
nev 



ne 



(2-47) 



All three quantities on the right-hand side of Eq. (2-46) can be measured, and 
so the Hall coefficient and the carrier density n can be found. 



<>y 



Face 2 




Direction of 

positive hole 

motion 



Face 1 
Direction of B out of paper 

FIG. 2-12. Motion of positive holes in a p-type semiconductor 
(Hall effect). 

Figure 2-12 shows the conditions that exist in a similar p-type specimen 
when the current is carried entirely by positive holes. If the conventional 
current flow is in the same direction, the deflection of the holes is down- 
wards, but now the carriers have a positive charge. Thus the bottom face 
becomes positive, and the potential difference and the field are in the opposite 
direction from the previous case. The sign of the charge and the direction of 
the field are both changed so Eq. (2-45) still applies, i.e., 

$ = Bv (2-45) 

The Hall coefficient for hole conduction is now positive and is given by 



R * = M 



Substituting J = pev where p is the positive hole density gives 

1 



R P = - 

pe 



(2-48) 



(2-49) 



64 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

Thus the sign of the Hall voltage indicates which carriers predominate in 
the material, and the magnitude of the potential difference may be used to 
calculate the density of charge carriers. 

The analysis given above applies only when the charge carriers are free of 
attractive forces in the energy bands and when they move with a steady drift 
velocity v. This is not the case in a semiconductor, and a computation of the 
average speed leads to the conclusion that 

Rn = =* = =^ (2-50) 



Sne 


ne 


3tt 

Zpe ~ 


1.18 
pe 



Rv = T- = — (2-51) 

Spe pe 

For «-type material, the conductivity is given by 



r n = nep 



and in /?-type material 



a p = pefx p 



Hence, * - £ - -£=£ (2-52) 

and ^ = G fe = Tji ^ 53 > 

Therefore, the mobility of the charge carriers may be found if the con- 
ductivity and the Hall coefficient are known. 



BIBLIOGRAPHY 

General 

Dekker, Adrianus J., Solid State Physics, Englewood Cliffs, N.J.: Prentice- 
Hall, Inc., 1957 

Middlebrook, R. D., An Introduction to Junction Transistor Theory, New 
York: John Wiley & Sons, Inc., 1957 

Spenke, Eberhard, Electronic Semiconductors, New York: McGraw-Hill 
Book Company, Inc., 1958 

Sproull, Robert L., Modern Physics, New York: John Wiley & Sons, Inc., 
1956 

Hall effect 

Dunlap, W. C, An Introduction to Semiconductors, New York: John Wiley & 
Sons, Inc., 1957 



-BIBLIOGRAPHY 65 

Kittel, Charles, Introduction to Solid State Physics, 2d ed., New York: John 
Wiley & Sons, Inc., 1956 

Shive, J. N., Properties, Physics and Design of Semiconductor Devices, New 
York: D. Van Nostrand Company, Inc., 1959 



PROBLEMS 

2-1 A metal has its Fermi level 8.95 ev above the bottom of the conduction 
band. Find the momentum of an electron at the Fermi level assuming 
the free mass of an electron applies. How many energy states per unit 
volume are present below the Fermi level ? 

2-2 What is the range in energy in multiples of kT over which the Fermi 
factor varies from 0.99 to 0.01 ? What is this energy range in electron 
volts at 300° K? 

2-3 What fundamental concepts are used in the derivation of the energy 
density of states in the conduction band of a metal as a function of 
energy ? 

2-4 Determine the average energy of an electron in the conduction band of 
a metal at 0° K as a function of the Fermi level. 

ans. (3/5) E F 

2-5 What is the percentage error introduced by making the approximation 
in Eq. (2-14) that E 1 — E F » kT (in particular, that 100 millielectron 
volts » 25.8 millielectron volts)? 

2-6 What are the significant differences between crystalline solid state 
materials and gases that necessitate the use of different statistical 
treatments (Fermi-Dirac for solids and Maxwell-Boltzmann for gases) 
in the evaluation of electrical phenomena? 

2-7 Recalculate Eq. (2-22) using m c and m v for the mass of an electron in 
the conduction band and a hole in the valence band respectively. If 
w c l m v =1.5, what is the displacement of the Fermi level from the 
center of the forbidden gap when T = 300° K? 

2-8 At 300° K, the width of the forbidden energy gap in germanium is 
0.72 ev and the intrinsic carrier density is 2.5 x 10 13 cm -3 . Calculate 
the value m c m v /m 2 , where m is the free mass of an electron. 

2-9 Carefully distinguish among intrinsic, n-type and/?-type semiconductor. 
Assuming m c = m v , what is the position of the Fermi level at 300° K 
(a) in intrinsic germanium (b) in «-type germanium where N d = 10 15 
cm -3 (c) inp-type germanium where 7V a = 10 13 cm -3 ? 



66 ELECTRONS AND HOLES IN SEMICONDUCTORS CHAP. 2 

2-10 Calculate the resistivity of intrinsic silicon at 300° K from the data of 
Table 1-2. 

2-1 1 Calculate the density of donor atoms which has to be added to intrinsic 
germanium to produce n-typ& material of conductivity 0.19 ohm cm. It 
is given that the mobility of electrons in the «-type semiconductor is 
3250 cm 2 volt" 1 cm" 1 . 

ans. 1.01 x 10 16 cm" 3 

2-12 Explain how the Hall effect shows whether holes or electrons pre- 
dominate in a semiconductor. Suggest an experimental arrangement 
for measuring Hall voltage giving values of current, magnetic field, and 
voltage you would expect to find for a small specimen of «-type ger- 
manium of resistivity 5 ohm cm. 

2-13 An w-type germanium specimen has a donor density of 10 15 cm" 3 . It 
is arranged in a Hall effect experiment where B = 0.5 webers m~ 2 and 
J = 500 amps m -2 . What is the Hall voltage if the specimen is 3 mm 
thick? 

ans. 5.5 millivolts 



3 



Junctions 
between Materials 



Before discussing semiconductor junctions, we will 
briefly consider the less complex case of a junction 
between two metals. It is shown that there is an 
initial passage of charge across the junction which 
causes a change in potential between the two sides. 
A semiconductor junction is then examined in more 
detail, and the contact potential, the junction 
capacitance, and the current versus voltage char- 
acteristic are derived. 



68 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



3-1 Work Function in a Metal 



Figure 3-1 shows the energy state density curve and the position of the Fermi 
level for a typical metal. The curve of S(E) versus energy is of the form 

S(E) = K(E) 1 ' 2 (3-1) 

where the energy, E, is measured from the bottom of the band, and K = 2 7/2 
m 3l2 7rh- 3 . 

The Fermi factor, F(E), has the value given in Eq. (2-10) which is re- 
peated here for convenience 

1 



F(E) = 



1 + exp [(E - E F )/kT] 




0°K 





S(E) —*■ F(E) — * N(E) 

FIG. 3-1. Position of the Fermi level for a metal. 



The curve of the number of occupied states, N(E), versus energy which is on 
the right-hand side of Fig. 3-1 is obtained from Eq. (2-11), i.e., 

N(E) dE = S(E)F(E) dE 

For a metal, the total number of conduction electrons remains constant 
as the temperature is changed. The variation of F(E) with temperature has 
been discussed in Chap. 2. However, it is found that, although the shape of 
the curve of Fermi factor plotted against energy is dependent upon tem- 
perature, the curve alters in such a way that the Fermi level is substantially 
constant. The Fermi level of a metal, therefore, is a useful reference level. 

In any metal the energy states at the top of the band are not completely 
full. Because there is a continuous band of energy states, electrons are free to 
move about within the metal, and the electrical conductivity is high. This 
leads to a picture of the potential energy variation in a metal which ignores 
the potential energy variations close to an atom but has the merit of simplicity. 
Sommerfeld first introduced this simple model of electrons in a metal as 



SEC. 3- 



WORK FUNCTION IN A METAL 



69 



given in Fig. 3-2. Here electron levels up to the Fermi level E F are shown 
filled. The vacuum level in the diagram refers to the energy of an electron at 
rest outside the metal. 

The vacuum level, E s , must be higher than the Fermi level, E F , for a 
metal at room temperatures, otherwise electrons would be thermionically 
emitted (see below). The difference between E s and E F is known as the work 
function, (/>, of the material. It is the energy required to raise one electron from 
the Fermi level inside the metal to the vacuum level outside. The value of the 
work function is usually between 2 and 5 ev. 

The Fermi level is measured from the bottom of the energy band and can 
be calculated for a particular metal if the valence, density, and atomic weight 
of the metal are known. The valence of a metal is the number of electrons per 
atom in the outer energy band, and so the product of the valence and the 
number of atoms per unit volume gives the total number of electrons per unit 




[— i 1 Vacuum level E c 



Bottom of energy band 
Metal 

Distance 



Vacuum 



FIG. 3-2. Simple picture of electrons in a metal. Conditions close 
to the surface of the metal are not shown. 



volume which are occupying energy states in this band. If this number is N, 
from Eqs. (3-1) and (2-11), 



N 



P KE ll2 F{E) dE = C F KE 112 dE 

Jo Jo 



(3-2) 



at T = 0° K, since F(E) = 1 up to E F and all states will be filled up to the 
Fermi level and unfilled above that level. Hence, 



N = 



2KEV 2 



or 



m 



2/3 



(3-3) 



Table 3-1 gives the values of the Fermi level and the work function for several 
metals. 



70 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



TABLE 3-1. Valence, Fermi level, and work function for 
five common metals. 



Metal 



Valence 



Al 


3 


11.7 ev 


4.20 ev 


Ag 


1 


5.5 


4.46 


Cu 


1 


7.0 


4.45 


K 


1 


2.1 


2.22 


Na 


1 


3.1 


2.28 



Electrons having an 
energy greater than 
the vacuum level E s 



As the temperature of a metal (or semiconductor) is raised, it is found 
that electrons will be emitted from the surface. This phenomenon is known 

as thermionic emission and is of great 
importance in thermionic vacuum 
tubes where it supplies the electron 
current in the device. The electron 
state distribution curve of Fig. 3-3 
shows that at high temperatures there 
is an appreciable number of electrons 
in the "tail" of the curve above the 
vacuum level, E s . This number can be 
calculated at any given temperature 
from Eq. (2-11). These electrons have 
sufficient energy to escape from the 
metal surface over the potential barrier 
<j> (see Fig. 3-2). However, all elec- 

trons having an energy greater than 

N(E) — * £ s w ill not necessarily be emitted from 

the surface. The calculation of the 
emitted current takes into account only 
those electrons which are traveling 
with a velocity normal to the surface 
sufficient to overcome the energy barrier of the work function, <f>. 

This example briefly indicates how the Fermi-Dirac statistics can be used 
to explain a well-known phenomenon in physical electronics. In the re- 
maining sections of this chapter, the Fermi level will be shown to be an 
important tool in the analysis of current flow across metals and semiconductor 
junctions. 




FIG. 3-3. Diagram showing electrons in 
the tail of the distribution curve which 
have an energy greater than E s . 



3-2 Junctions between Metals 



In Fig. 3-4 is shown the energy band diagram at 0° K of two different metals 
having work functions fa and fa. The shaded areas represent the filled 



SEC. 3-2 JUNCTIONS BETWEEN METALS 

Metal 1 



Filled electron 
levels 




E* 



(Vacuum level) 



Fermi level 



Fermi level 



Metal 2 




71 



Filled electron 
levels 



+- N(E) 



N(E) 



FIG. 3-4. Energy band diagram of two metals with work func- 
tions (f>i and <f> 2 (metals are not in contact). 



Metal 1 



E, 



Transition 
region 



01-02 



Filled electron 
levels 




Fermi level 



Metal 2 



N(E) 



Surface potential 




Filled electron 
levels 



ME) — 



FIG. 3-5. Junctions of two metals at 0° K showing the contact 
potential (metal 1 at ground potential). 



72 JUNCTIONS BETWEEN MATERIALS CHAP. 3 

electron levels in each metal, and the metals are assumed to be separated. 
The reference level E s is common. 

Figure 3-5 shows the energy band diagram when the two metals have a 
common junction. The junction is assumed to be clean and regular so that 
the crystalline properties of the two materials are altered as little as possible. 
Since the electrons are virtually free in both metals, there is a transfer of 
electrons from metal 2 to metal 1 until the Fermi levels coincide. In this 
process, metal 1 acquires electrons and becomes negative, and metal 2 loses 
electrons and becomes positive. The difference in energy possessed by an 
electron on the two sides of the junction is <j> x — <f> 2 = eV B . Thus V B is the 
potential difference between the two metals which occurs because of the 
transfer of charge across the junction. V B is known as the contact potential 
between the metals. 

For metals at 0° K it can easily be seen that, since the highest filled 
electron states coincide with the Fermi level, the two Fermi levels on either 
side of the junction will line up. At any other temperature, there are some 
filled electron states above the Fermi levels, but it may be shown thermo- 
dynamically that, at any temperature, the Fermi levels of any two materials 
will coincide when the two materials have a common junction. 

3-3 Semiconductor Junctions with No Applied Voltage 

Semiconductor junctions are commonly formed between p- and «-type 
material. It is assumed that the group 4 crystalline structure is unaltered 
across the junction and that only the type of impurity changes. Thus, 
junctions having the properties described here cannot be made by placing 
two pieces of semiconducting material together, because surface films and 
other irregularities would produce major discontinuities in the crystal 
structure. Junctions can be made by diffusing the required impurity density 
into the crystal, by an alloying process, or by growing a crystal with an 
impurity density which is a function of distance. These three methods of 
construction are briefly discussed in Sec. 5-6. 

Junctions may be abrupt or graded. An abrupt junction, which is also 
known as a step junction, is of the type shown later in Fig. 3-7 where the 
acceptor or donor density is a constant in the material up to the boundary. 
In a graded junction, the impurity density varies with distance away from the 
boundary in some manner. Both types of junction are used extensively in the 
manufacture of diodes and transistors. Only the abrupt type of junction will 
be discussed in detail in this chapter. 

In a metal, the electrons which move across the boundary to bring the 
Fermi levels in line are drawn from a very thin layer close to the junction 
since there is a high density of free electrons in the metal (about 10 22 electrons 
per cubic centimeter). For a semiconductor, however, there may be only 
about 10 15 free electrons per cubic centimeter, in a typical case, and so there 



SEC. 3-3 



SEMICONDUCTOR JUNCTIONS WITH NO APPLIED VOLTAGE 



73 



is a much wider region which is affected by the flow of charge. Electrons in 
the conduction band on the n-typt side, which were supplied from the group 5 
impurity atoms, travel across the junction and leave the positively ionized 
group 5 atoms unneutralized (Fig. 3-6). Consequently there is a positively 



p-type 



rc-type 




Metallurgical 
junction 

Distance - 



p-type 



Group 3 atoms 
shown only-. 



n-type 

Metallurgical ~, - , 

, . Group 5 atoms 

junction shQwn only 



o o © © 

b °o © © 

D O 

o o © © 



© © o o 
© © o* o # 
© © o # o* 

}_ 



Positive hole 

and group 3 

atom (neutral) 



Extent of 
negative 

space-charge 
region on 

p-type side 

Charge 

depletion 

region 



Extent of I f 

positive J Electron and 
space-chargei group 5 atom 
region on ] (neutral) 
rc-type side | 

Charge i 

depletion j 

region \ 



FIG. 3-6. The p-n junction. 



charged region adjacent to the junction in the /?-type material. On the /?-type 
side, the electrons which have traversed the boundary recombine with the 
positive holes in the valence band, which predominate because of the pres- 
ence of group 3 atoms. Close to the junction on the /?-type side, there is a 
layer of unneutralized negatively ionized group 3 ions which form a negatively 
charged region. Both the group 5 positive ions (in the «-type material) and 
the group 3 negative ions (in the /?-type material) are immobile since they are 
bound in the crystal lattice. Close to the metallurgical junction, therefore, 
there are very few mobile charge carriers, and this location is referred to as the 
charge depletion region. In fact, it is a charged or space charge region because 
of the presence of the immobile ions, but there are no charges available for 
conduction. 

Figure 3-7(b) shows the two charge depletion regions of opposite sign on 
the two sides of the junction. Since the numerical values of the charge on the 
two types of ion are the same, for overall charge neutrality 



N d X 2 = N a X 1 



(3-4) 



The charge depletion region in each material is therefore inversely propor- 
tional to the donor or acceptor density in the material. The effective width of 
the junction (that is, the region over which the bulk properties of the p- or 
«-type material no longer apply) is X 1 + X 2 . This quantity is of considerable 
importance in junction transistors as will be shown later. Also of interest in 



74 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



both p-n junctions and transistors is the capacitance across the junction 
because of the two space charge regions (see Sec. 3-4). 

There is an important relationship, known as Poissorfs equation, which 
may be used to find the way in which the potential varies with distance in a 



p-type a 

S T3 


n- 


type 






N 





(a) 



Negative 
charge 




FIG. 3-7. (a) Impurity density, (b) Charge density, (c) Potential 
across a p-n junction. 

region containing charge. In three dimensions, and written in nonvectorial 
form, Poisson's equation is 

8 2 V d 2 V d 2 V _ -p 

8x 2 + dy 2 + dz 2 ~ e 

where p is the volume charge density in the material of permittivity e. 



S€C. 3-3 SEMICONDUCTOR JUNCTIONS WITH NO APPLIED VOLTAGE 75 

In the more simple one-dimensional case that we are using in this analysis, 
Poisson's equation reduces to 

In the space charge region of the /?-type material in the junction shown in 
Fig. 3-7 and assuming that the junction has unit area, 

P= -eN a (3-6) 

TU d 2 V -p eN a _. 

Thus, _ = _ = __ (3-7) 

Integrating, 

dV eN 

^- = — x + C (3-8) 

dx s 

eN x 2 
and V = A^- + Cx + D (3-9) 

Equation (3-9) applies only in the space charge region of the p-type 
material. To evaluate the constants, C and D, we will investigate the bound- 
ary conditions at the two ends of the region. At the right-hand side of this 
region in Fig. 3-7, we can write V = when x — 0, if all voltages are 
measured by reference to the potential at the metallurgical boundary of the 
two materials. In the bulk/?-type semiconducting material, we know that the 
potential is uniform. Therefore, in Fig. 3-7, the space charge region can be 
said to end at a point x = — X ± where dV/dx = 0. Inserting these two 
boundary conditions into Eqs. (3-9) and (3-8), respectively, 

D = 
from Eq. (3-9), and 

eN 

£ 

from Eq. (3-8). If V = V ± when x = -X l9 

Vl = Z£^£ xl (3-10) 

Is 

If Poisson's equation for the n-typc depletion region is considered, 

and by a similar argument it can be shown that 

V 2 = 6 -^X\ (3-11) 

where V = V 2 at x = X 2 (the limit of the n-typc depletion region in Fig. 3-7). 



76 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



The total potential across the junction when no external voltage is sup- 
plied is the contact potential, V B . 



V B = |Fi| + \V 2 \ = T£ {N a X\ + N d Xl) 

From charge neutrality, 

N a X ± = N d X 2 

Eliminating X 2 from Eqs. (3-12) and (3-4), 

v = eN a X\ I, , N a 



(3-12) 

(3-4) 



or X x = 

Alternatively, if X 1 is eliminated, 

Xo. = 




1/2 



(3-13) 



1/2 



(3-14) 



Equations (3-13) and (3-14) show that the depletion widths can be calculated 
from a knowledge of the contact potential and the donor and acceptor 
densities in the n- and /?-type semiconducting regions. 

It is possible to calculate the contact potential if N d and N a are given for 
the two regions. For the junction of Fig. 3-7, we can write n n for the equilib- 
rium density of electrons in the rc-type bulk material (away from the space 
charge region), and n p for the equilibrium density of electrons in the /?-type 
bulk material (away from the space charge region). Then n n = N d where N d 
refers to the «-type material, and n p = nf/N a where N a refers to the p-typc 
material, and n p is obtained from the law of mass action [Eq. (2-34)]. In 
Fig. 3-8, we have repeated the energy band diagram of the p-n junction. The 
Fermi levels on the two sides of the junction are coincident, and the diagram 
also indicates the relative positions of the Fermi levels of intrinsic material 
with reference to the energy gap. If E p and E n are the energy levels of the 
bottom of the conduction band on the /?-type and «-type side respectively, 
we can write 

E p - E n = eV B (3-15) 

In addition, from Eqs. (2-23) and (2-25), 

n n = N d = tiiCxp^J 



and 



nf 



m 



SEC. 3-3 



SEMICONDUCTOR JUNCTIONS WITH NO APPLIED VOLTAGE 



77 



where AE and AE' are as defined in Sees. 2-4 and 2-5 and as shown in Fig 
3-8. From the diagram, 

AE + AE' = eV B 
Thus, 



or 



n n (AE + AE'\ (eV B \ 



'-?'* ©-¥■*(¥) 



(3-16) 
(3-17) 
(3-18) 



<D 



Fermi level of 
intrinsic material 
Fermi level 
(p-type) 



p-type 


n-type 


n p 










n n 


Jae' 






Jae 



















Conduction 
band 



Fermi level (n-type) 

- Fermi level of 
intrinsic material 



Valence 
band 



Metallurgical 
junction 

Distance — ► 
FIG. 3-8. Energy band diagram for the p-n junction. 

Example : 

A p-n junction is formed from germanium of resistivity 1 ohm cm (p side) and 
0.1 ohm cm (n side). When operating at room temperature (300° K), what is 
the voltage across the junction (contact potential), and what is the total width 
of the depletion region? [Note: Many of the quantities required in this cal- 
culation are commonly given in c.g.s. units whereas most electrical calcula- 
tions are made using practical (M.K.S.) units. Familiarity must be obtained 
in converting from one set of units to the other.] 

Since the resistivity of both sides is considerably less than the resistivity of 
intrinsic germanium, 7V d » n { (w-type) and /V a » n { (p-type). For the «-type 
material where a n = 10 mho cm -1 , a n = N d e^ n and the electron mobility, 
/Lt n , from Fig. 1-21 is 3000 cm 2 volt -1 sec -1 . Hence, 



JV„«-3l 



10 



en n (1.6 x 10" 19 )3000 



« 2.1 x 10 16 cm" 3 



2.1 x 10 22 m" 3 



78 JUNCTIONS BETWEEN MATERIALS CHAP. 3 

For the p-type material, the hole mobility, \i v , from Fig. 1-22 is 1650 cm 2 
volt -1 sec -1 when a p = 1 mho cm" 1 , and 

AT = —L = : = 18 x 10 15 cm -3 

" a ep p (1.6 x 10" 19 )1650 ** X 1U Cm 

= 3.8 x 10 21 m- 3 

At a temperature of 300° K, 

kT (1.38 x 10- 23 )300 



e " 1.6 x 10" 19 



25.8 mv 



Hence, 



v kT. (N d N a \ 

_ c , [(2.1 x 10 16 )3.8 x 10 15 1 
= 25 ' 8 l0ge I 6.25 x 10 2 ° J 



From Eq. (3-13), 



= 303 mv 



*i = 



2eV f 



eN c 



(>♦» 



1/2 



Working in M.K.S. units and noting that the relative permittivity of 
germanium is 16, 

_ [ 2(16)(8.85 x 10- 12 )0.303 I 1 ' 2 
Al ~ Ll.6 x 10" 19 (3.8 x 10 21 )(1 + 0.183)J ~ 5A:> x 1U m 

Similarly, from Eq. (3-14), 

_ [ 2(16)(8.85 x 10- 12 )0.303 V' 2 _ 
* 2 ~ Ll.6 x 10" 19 (2.1 x 10 22 )(1 + 5.53)J " 6 ' 23 x 1U m 

Total depletion width = |*i| + \X 2 \ = 4.08 x 10" 7 m 

Because of the contact potential across a p-n junction, the energy of an 
electron at the bottom of the conduction band in the p-type material is 
greater than that of a corresponding electron at the bottom of the conduction 
band in the /7-type material by an amount eV B (Fig. 3-8). The presence of the 
potential across the p-n junction, therefore, can be thought of as a potential 
barrier existing between the two sides. The voltage, V B , is in such a direction 
as to present a potential barrier to electrons which try to move from the n to 
the p side, whereas, it will assist electrons moving from the p to the n side. 
For an open-circuited junction, the net flow of current must be zero, and this 
condition is used in the analysis of the junction in Sec. 3-5. 



SEC. 3-4 



JUNCTION WITH APPLIED VOLTAGE: JUNCTION CAPACITANCE 



79 



3-4 Semiconductor Junction 

with Applied Voltage: Junction Capacitance 

When a voltage is applied to a p-n semiconductor junction, it is found that 
the voltage-current characteristic is nonlinear. If the p-type material is made 
positive with respect to the ft-type, the junction is said to be forward biased 





p-type 


n-type 


-i 




=ft* 





Forward bias 



Forward bias 



p-type 


n-type 



Reverse bias 



7 



^ 



+ v 



Reverse bias 

FIG. 3-9. The p-n junction and its characteristics under forward 
and reverse bias. 

since this is the direction in which the resistance of the device is low and the 
current flows more readily. When the «-type is made positive with respect to 
the /?-type, the junction is reverse biased, and its resistance is high (Fig. 3-9). 



p-type 



n-type 



Fermi level %¥ (Forward 

^ T bias) 



Distance 



FIG. 3-10. A p-n junction with forward bias. 



This rectifying property of the p-n junction will later be explained in terms 
of the previously developed theory of semiconductors. 

When only a small current is flowing in a typical p-n junction, the voltage 
drop across the p- or «-type bulk material is usually small compared with the 



80 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



voltage drop across the junction. In the analysis that follows, therefore, the 
voltage drop in the semiconducting material itself will be neglected, and all 
the applied voltage will be assumed to be dropped across the junction. 

The presence of an applied voltage across the junction adds to, or sub- 
tracts from, the barrier voltage V B . Consider the p-n junction with forward 
bias as shown in Fig. 3-10. When an electron passes across the junction from 
the n- to the p-type material, it is moving in the same direction as the force 
due to the applied electric field. Thus the potential energy in the />-type 
material is less than it would have been in an unbiased junction by an amount 
V electron volts, and in Fig. 3-10, the conduction and valence bands have 
been lowered by this value. (Compare Fig. 3-10 with the energy bands for 
the unbiased junction given in Fig. 3-8.) The energy band structure of the 



p-type 



n-type 







x^ 




a 


t 


Fermi level 

Reve 


i\_ 




I 


rse bias eV 

< r 


Fermi level 


s 


a 




2 

> 

•"tt 
O 




J 

Barrier v 
height eVT 






I 



Distance 



FIG. 3-11. A p-n junction with reverse bias. 

p-type material is unchanged, however, and so the Fermi level, too, is 

lowered V electron volts with respect to its counterpart in the n-type material. 

Since the unbiased barrier voltage (the contact potential V B ) is reduced by 

a forward bias V, the barrier voltage in the general case is given by V T where 

V T = V B - V (3-19) 

Figure 3-1 1 illustrates the case of the p-n junction with reverse bias. The 
barrier height is now increased over the unbiased value. 

Referring back to the analysis of the preceding section, Eqs. (3-4) to 
(3-14) still apply if the total barrier voltage V T is substituted in place of the 
unbiased value V B . For example, Eq. (3-13) can be rewritten as 



Xi = 



leV, 



eN D 



(■♦a 



(3-20) 



SEC. 3-4 JUNCTION WITH APPLIED VOLTAGE: JUNCTION CAPACITANCE 81 

where X x is now the depletion width in the p-type material with an applied 
voltage V. Similarly, 

2sV T 



Xo = 



eK 



(-3 



11/2 

(3-21) 



where X 2 is the depletion width in the «-type material with applied voltage V. 
Each space charge region of the junction contains a charge which is 
numerically equal to Q, where 

\n\ *NY .NY \ ^sN a N d V T y 12 

\Q\=eN a X 1 = eN d X 2 =[ {Na + Nd) \ 

from Eqs. (3-20) and (3-21) and from Fig. 3-7(b). Equation (3-19) shows 
that, when a forward voltage V is applied, V T decreases, and Q is diminished. 
From Fig. 3-7(b), the «-type region contains positive charge, and the /?-type 
region contains negative charge. These charges are both of the opposite sign 
to the charges on an equivalent capacitor when a forward bias is applied. 
Thus the junction charge, Q, should be written with a negative sign to give 

= _\ 2eeN a N d V T y» (3 _ 22) 

The variation in junction charge, Q, as the applied voltage, V, is altered 
constitutes an incremental junction capacitance, C, which is given by C = 
dQIdV. From Eq. (3-19), 

dV T = -dV 

and so, by differentiation of Eq. (3-22), 

dQ 



C = dV 



ctiy a iy d 17-1/2 H-2TI 

2(N a + N d )\ { " l5) 



By an algebraic manipulation of Eqs. (3-20), (3-21), and (3-23), we can 
write, 

c =xnrm (3 - 24) 

The last two equations show that across an abrupt semiconductor junction, 
there is a nonlinear incremental capacitance of value equal to the quotient of 
the permittivity of the medium and the total depletion width. For the 
junction used as an example in Sec. 3-3, 

|JSTi| + \X 2 \ = 4.08 x 10" 7 m 

and, taking the relative permittivity of germanium as 16, the capacitance per 
unit area is 

r g 16(8.85 x 10" 12 ) _ f _ 2 

C = m + 1^ 1 = 4.08 x 10-^ = 346 ^ fm 



82 



JUNCTIONS BETWEEN MATERIALS 



CHAP. 3 



For a junction of area 10 ~ 6 m 2 , G = 346 pf. This is the capacitance across 
the junction when the applied voltage is small compared with V B . In practice, 
the capacitance is more frequently of concern when the p-n junction is 
reverse biased, since then its reactance may be smaller than the resistance of 
the junction in the reverse direction. If the applied voltage is - 10 volts, 

V T = V B - V = 0.303 + 10 = 10.3 volts 

The capacitance of the junction then becomes 59 pf. Note from Eqs. (3-23) 

and (3-24) that if one of the regions is 



p-type 



n-type 



of high conductivity, it has little effect 
on the junction capacitance which is 
then determined by the conductivity, 
i.e., the doping, of the other region. 

For a linearly graded junction 
where the donor and acceptor densities 
are as given in Fig. 3-12, the incre- 
mental junction capacitance is pro- 
portional to Vt 113 . The relationship 
between capacitance and voltage can 
be derived in a similar way to the 
analysis for the abrupt junction (see 

problem 3-4). The nonlinear capacitance versus voltage characteristic is 

useful in some types of electronic circuitry. 




FIG. 3-12. A linearly graded junction 
where N d — N a = ax and a is a constant. 



3-5 Flow of Current 

across a p-n Junction: The Rectifier Equation 

In this section, we shall quantitatively discuss the flow of current across a p-n 
junction and derive the form of the rectifier equation. This will give a physical 
picture of the semiconductor rectifier before the more exact analysis is given 
in Chap. 4. 

Consider Fig. 3-13 which shows the energy bands of a p-n junction. In 
the diagram, a forward bias has been applied, and so the Fermi level on the 
ft-type side, E Fn , is higher than that on the p-type side by the value eV. 
There are four possibilities for current flow across the junction, and these 
have been shown in the figure. In the valence band, holes can flow to the 
right or to the left, and the two current densities have been separately 
identified as Jf and J 2 + respectively. In the conduction band, electrons can 
flow to the right or to the left. A flow of electrons to the left produces a 
conventional current flow to the right since electrons carry a negative charge. 
For this reason, the arrows in the conduction band on Fig. 3-13 indicate the 
direction of flow of the electrons, and a minus sign has been placed before the 
current densities. Thus current density J± is flowing to the right, and current 
density J 2 is flowing to the left. This means that the two current densities 7r 



SEC. 3-5 FLOW OF CURRENT ACROSS A p-n JUNCTION: THE RECTIFIER EQUATION 83 

and J± add together in the right-hand direction and current densities / 2 ~ and 
J} add together in the left-hand direction. The net current density flowing 
toward the right is 

J = (/£■ + J?) - (J* + J}) (3-25) 

The variation of all four current densities with applied voltage, V, will now be 
discussed. 

In the conduction band, current density J^ is caused by a flow of electrons 
from the /?-type material down the potential hill into the fl-type. In a typical 
junction, this is a small current of a few microamps since its supply of charge 

p-type n-type 



• E F Fermi level 



2 Fermi level 



Distance — *■ 

FIG. 3-13. A p-n junction showing electron and hole current 
densities. (Note that the arrows show the direction of flow of 
both electrons and holes. For J{ and / 2 ~ the flow of conventional 
current is in the opposite direction.) 

is the conduction band of the p-typQ material where electrons are scarce. The 
mechanism of the transfer of charge from the bulk material to the depletion 
region is mainly that of diffusion which will be discussed in Chap. 4. Here, we 
note only that the passage of electrons into the depletion region by diffusion 
is nearly independent of the voltage across the junction since the two regions 
are substantially independent of one another. Once an electron is at the edge 
of the potential barrier (coordinate — X x on Fig. 3-7), it will be swept across 
the junction by the action of the electric field. Changing the height of the 
barrier will not alter /£ provided the barrier voltage remains in the sense 
shown in Fig. 3-13. In other words, J% will be constant provided the electrons 
flow down a potential hill. This condition is true for all values of reverse bias, 
and also when the forward bias is less than V B . J^ is often known as a 
saturation current density for this reason. 

Ji is the other current density in the conduction band, and here electrons 
flow from the «-type to the /?-type semiconductor. The conduction band of 



84 JUNCTIONS BETWEEN MATERIALS CHAP. 3 

the H-type material provides a copious source of electrons for this current. 
Provided the current is not excessive, the only restriction on the flow of the 
current is that of the presence of the barrier potential V T . The probability of 
occupation of the states in the conduction band of the «-type material 
decreases exponentially with increasing energy in the band according to 
exp [(E Fn — E)/kT]. For a potential barrier of height V T above the bottom 
of the conduction band of the «-type material (as shown in Fig. 3-13), the 
number of electrons which are an available source of current is proportional 
to exp (-eV T /kT). Thus the flow of electrons from right to left across the 
barrier causes a current 

Ji = C, exp (^ffj (3-26) 

where C x is a constant to be determined. 

A similar argument may be used to find the two positive hole current 
densities in the valence band. J% is a saturation current density which does 
not depend on the height of the potential barrier. Since holes carry a positive 
sign, holes flowing upwards in Fig. 3-13 are effectively flowing down the 
potential barrier. Current density /J", which is flowing in the opposite 
direction, is moving up the potential barrier and is given by 

y x + = C 2 exp(^) (3-27) 

where C 2 is another constant. 

There is no net current flow across the junction when the applied voltage, 
V, is zero. From Eq. (3-19), V T = V B , and 

J- _ /- = (3-28) 

and J} -J£=0 (3-29) 

because current flow in one band is independent of the flow in the other band 
and charge cannot be accumulated in either band. Substituting for J x from 
Eq. (3-26) in Eq. (3-28) and using V T = V B 

c^p(=ffi)-j; = o 

d = Ji exp (^?) (3-30) 



or 
Similarly, 



C 2 = Ji exp (§) (3-31) 

Equations (3-26) and (3-30) relate Jf and J% in terms of V T and V B . Similarly, 
Eqs. (3-27) and (3-31) give the relationship between J} and J%. For the 



SEC. 3-5 FLOW OF CURRENT ACROSS A p-n JUNCTION: THE RECTIFIER EQUATION 



85 



general case of V ^ 0, we may substitute for J x and J± in Eq. (3-25), to give 
J = (Ji + J?) - W + JZ) 

= (/ 2 " +/ 2 + )[exp(g)- l] 

= /2 [ eXP ©" 1 ] (3 " 32) 

where J 2 = J* + «^2 + (the sum of the saturation current densities), and 
V = V B - V T from Eq. (3-19). 



AkT/e 



Positive quadrant 
forward bias 



J2 ex P (f|) 



+ V 



K 



4kT/e 



N - Breakdown 
region 



Negative quadrant 

reverse bias _ j 



FIG. 3-14. The voltage-current characteristic of a p-n junction 
rectifier. 

Equation (3-32) is known as the rectifier equation and is plotted in Fig. 
3-14. In the positive quadrant, when a forward bias is applied of magnitude 
eV > 4kT, Qxp(eV/kT) » 1, and J = J 2 exp (eV/kT). In the negative 
quadrant, when a reverse bias of magnitude — eV > AkT is applied, exp 
(eV/kT) « 1, and / = J 2 . At room temperature (T = 300° K), 4kT = 103 
millielectron volts. 

When the junction is biased in the forward direction, the potential barrier, 
V T , is lowered, and the currents Jl and J± are increased. For the reverse- 
biased junction, the potential barrier is increased, and the saturation current 



86 JUNCTIONS BETWEEN MATERIALS CHAP. 3 

densities J% and J} dominate the expression. The value of the saturation 
current is derived in Chap. 4. 

As the forward current increases, the characteristic in Fig. 3-14 becomes 
more linear. This is because the ohmic resistance of the semiconductor (which 
has so far been neglected) causes a significant voltage drop to occur in the 
body of the semiconductor. 

3-6 The Breakdown Region of a p-n Junction 

As the voltage is increased in the negative direction in Fig. 3-14, we come to a 
part of the curve labelled "breakdown region." Here, the current increases 
very rapidly. This is the effect of avalanche breakdown. An electron acquires 
sufficient energy in its passage in the electric field that it can excite an electron 
from the valence band to the conduction band, and so add an electron-hole 
pair to the existing charge carriers. The new charge carriers now take place in 
a secondary collision process of the same nature, and an avalanche of charges 
is obtained. This mechanism is very similar to the Townsend avalanche 
which is of great importance in gas discharge phenomena. 

The name "Zener breakdown" has been widely used to describe the 
breakdown region of the p-n rectifier characteristic and is still in use even 
though the avalanche mechanism is usually operative. Zener breakdown 
occurs when the electric field is so high that electrons are pulled out of their 
covalent bonds in a semiconductor. For most p-n junctions, avalanche 
breakdown will occur first. The avalanche breakdown voltage is very 
stable and forms a useful reference voltage in electronic circuitry. Semi- 
conductor manufacturers produce special devices for this purpose known as 
"Zener" (a misnomer), voltage regulator, or avalanche diodes. 

BIBLIOGRAPHY 

Greiner, R. A., Semiconductor Devices and Applications, New York: McGraw- 
Hill Book Company, Inc., 1961 

Middlebrook, R. D., An Introduction to Junction Transistor Theory, New 
York: John Wiley & Sons, Inc., 1957 

Shockley, William, Electrons and Holes in Semiconductors, New York: D. 
Van Nostrand Company, Inc., 1950 

Smith, R. A., Semiconductors, London: Cambridge University Press, 1959 

Spangenburg, Karl R., Fundamentals of Electron Devices, New York: 
McGraw-Hill Book Company, Inc., 1957 

Sproull, Robert L., Modern Physics, New York: John Wiley & Sons, Inc., 
1956 



PROBLEMS 87 

PROBLEMS 

3-1 Explain the terms (a) valence, (b) work function, (c) contact potential 
with reference to metals or metal junctions. 

3-2 An abrupt p-n junction is made of silicon where the resistivities of the 
two sides are 2 ohm cm (/7-side) and 1 ohm cm («-side). Using the in- 
formation given in Sees. 1-18 and 3-3 compute the contact potential 
and the total width of the depletion regions for zero applied voltage. 

ans. 0-30 volts, 3.1 x 10" 7 m 

3-3 Using the example in Sec. 3-3, find the positions of the Fermi levels on 
the two sides of the junction at 25° K. Using Eq. (2-35) and assuming 
that the exponential term dominates the expression, show that the 
variation in Fermi level with temperature for the «-type material may 
be calculated. From this argument, show that the contact potential 
decreases to zero as the temperature is raised. 

3-4 Show that the incremental capacitance of a linearly graded junction is 
given by 

_ (ea£_y> 3 
" \12V T J 

where N d — N a = ax and a is a constant (see Fig. 3-12). 

3-5 What is the incremental junction capacitance of a germanium p-n 
junction of area 10 ~ 3 cm 2 when a reverse voltage of 1 volt is applied? 
The conductivity of the «-type side is 1 mho cm -1 and that of the/?-type 
side is 2 mho cm -1 . 

3-6 What is meant by a space charge region? Discuss the variation in the 
space charge region of an abrupt p-n junction which occurs when the 
applied voltage is changed. Show that the junction has an incremental 
capacitance and suggest a use for this nonlinear capacitance. 

3-7 Show that for the condition where the acceptor density in the /?-type 
region is greater than the donor density in the «-type region, the total 
depletion width of the p-n junction when there is no applied voltage is 



m 



3-8 Find the depletion widths and the junction capacitance for the p-n 
junction used as an example in Sees. 3-3 and 3-4 when (a) a reverse bias 
of 4 volts is applied (b) when a forward bias of 0.04 volts is applied. 

3-9 Describe the four currents which may be considered to flow across the 
p-n junction. What is meant by the saturation current density? Show 
that for very small values of applied voltage (whether forward or 



88 JUNCTIONS BETWEEN MATERIALS CHAP. 3 

reverse) the p-n junction behaves as a small linear resistance of value 
0.026// 2 ohms at room temperature, where I 2 is the total saturation 
current across the junction. 

3-10 A p-n junction is constructed with conductivities of 200 mhos m _1 
(«-side) and 500 mhos m -1 (/?-side). The total length of the device is 
2 mm and the cross-sectional area is 1 sq mm. The saturation current in 
the reverse direction is 2 microamps at room temperature. Compute the 
voltage across the whole device when the current is 10 ma assuming the 
junction is at the center of the device and that no temperature rise takes 
place. 



4 



The It has been shown that the number of free electrons 
and positive holes available for the passage of 
Conti nu ity current in a semiconductor is considerably less than 
the number of free electrons in a metal. This fact is 
equation f importance in analyzing the flow of current in 
semiconducting materials since the equilibrium 
charge density is disturbed by the passage of a 
comparatively small current density. In this chap- 
ter, a fundamental equation is developed which 
governs the behavior of charge carriers in the 
material. This is known as the continuity equation. 



90 



THE CONTINUITY EQUATION CHAP. 4 



4-1 Derivation of the Continuity Equation 

The analysis of the charge distributions and current flow in a piece of semi- 
conducting material will be carried out by taking into account the effects of 
three phenomena. These are : 

1 . the generation and recombination of electrons and positive holes, 

2. the drift of charge in an electric field, 

3. the diffusion of charge as a result of a charge concentration gradient. 

Among the other effects which may have to be considered in particular cases 
are temperature gradients and applied magnetic fields. However, in most of 
the problems which will be encountered in practice it is necessary only to 
consider the three most important phenomena listed above. 

The continuity equation is formed by adding the three rates of delivery or 
removal of charge in a given volume of the semiconductor and equating this 



dx 



Unit area 



■^J+dJ 



dx 



h 



FIG. 4-1. Volume element used in the derivation of the con- 
tinuity equation. 

total to the rate of change of charge density in the material. This computa- 
tion is a more sophisticated version of the calculation of the change in height 
of liquid in a tub when water is entering through an inlet and leaving through 
an outlet. The semiconductor problem is complicated by having two charge 
carriers of opposite sign (negative electrons and positive holes) and by the 
possibility of the presence of a nonuniform charge density through the 
material. In practice, two continuity equations may be established: one will 
be for electrons, and one will be for positive holes. 

The continuity equation is extremely important in the understanding of 
the properties of semiconductor diodes and transistors, and it will be used 
extensively in this book. For simplicity, the one-dimensional form of the 
equation will be derived here. Although more complete, the three-dimensional 
form is more difficult to solve and gives little additional information for our 
purpose. 

The volume element in the semiconductor which is to be used in the 
derivation of the continuity equation is shown in Fig. 4-1. It is a three- 
dimensional element of unit area in the yz plane and of length dx in the x 



SEC. 4-1 DERIVATION OF THE CONTINUITY EQUATION 91 

direction. The total volume of the element is dx. The continuity equation will 
first be derived for positive holes only. 

Let the average density of positive holes in the volume element at any 
time be p per unit volume. Restricting the argument to the one-dimensional 
case, assume that the current density entering the element by the left-hand 
face is /, and the current density leaving the right-hand face is J + dJ. The 
difference between these two current densities shows that dJ/e positive holes 
are being removed from the volume element every second by current flow. 
Note that this lack of continuity in current indicates that other mechanisms 
are operating in the delivery, removal, or storage of charge in the element. 

The three phenomena which affect the charge density in the semiconductor 
will be discussed in the order in which they were listed. 

1. Generation and recombination of electrons and positive holes. It was shown in 
Sec. 1-20 that the rate of change of electron density with time could be 
represented by 

dn _ n — n 
dt " r n 

where the generation rate, « / T n, and the recombination rate, n/r nt were the 
only factors affecting the electron density. 
By a similar argument, 

dp = P^P (4 _, } 

dt r p 

where p is the equilibrium density of positive holes in the material, p is the 
positive hole density at any time, and t p is the hole lifetime. This equation 
determines the change in hole density because of generation and recombina- 
tion. 

2. Drift of charge in an electric field. Suppose an electric field, £, is applied 
along the x axis of the volume element. Positive holes will flow in the x 
direction with an average velocity v where 

v = n P * (4-2) 

and jjL p is the mobility of positive holes in the material. This flow of charge 
constitutes a current density in the x direction of magnitude 

J£ T = pev = pe^pfi (4-3) 

where the superscript + indicates that this is a current of positive holes, and 
the subscript Dr stands for drift to distinguish it from the case of diffusion. 
Equation (4-3) will be used later when the terms in the continuity equation are 
assembled. 

3. Diffusion of charge. The phenomenon of diffusion plays an important role 
in transistor operation. Diffusion is a process whereby charge carriers (holes 



92 THE CONTINUITY EQUATION CHAP. 4 

or electrons) can move in a material independently of the electric field. Dif- 
fusion is the flow of carriers from regions of high carrier density to regions of 
low density and, therefore, occurs only where there is nonuniform carrier 
density in the material. As an illustration, assume that the hole density at one 
point in a semiconductor specimen is increased by some means. There will 
then be a flow of holes from the high to the low density region so that the hole 
density becomes more uniform in the material. If the mechanism which 
produced the excess holes stops operating, the effect of diffusion is to bring 
the hole density back to uniformity in the specimen. At all times, diffusion is 
a flow of carriers in the direction of decreasing carrier concentration regard- 
less of the sign of the charge on the carrier. 

We can represent the diffusion of holes in a semiconductor by the equation, 

N--.D.% (4-4) 

In this equation we are postulating a change in hole density in the x direction 
along the volume element and stating that TV is the number of holes diffusing 
per unit area in the x direction per second. The minus sign is included in the 
equation because the diffusion of holes is in the direction of decreasing 
concentration, that is, TV will be positive when dp/dx is negative. The diffusion 
constant D p is a constant for positive holes in a particular material. A 
typical value for D p is 49 cm 2 sec -1 for intrinsic germanium (see Table 1-2). 
The current density carried by the diffusing holes is 

J£ i = eN= -eD p ^ (4-5) 

The subscript Di indicates diffusion and the superscript + shows that the 
current is carried by positive holes. Here the negative sign shows that the 
current is diffusing away from the region of maximum concentration density. 

The sum of the drift and diffusion currents must be taken into account in 
determining the rate of change of charge density with time. The total current 
density of positive holes is 

j + =/ D + r + y D + t (4-6) 

and the net flow of current from the element is dJ (see Fig. 4-1). 

Considering only the phenomena of drift and diffusion, the charge 
density may be postulated to change at the rate of dpjdt. Thus the rate of 
increase of the total number of positive holes in the element is 

¥t dx 
since dx is the volume of the element. If the net current flow is responsible for 



SEC. 4-1 DERIVATION OF THE CONTINUITY EQUATION 93 

this change in holes contained in the element, 

dt ax e 

or -£ = - - — (4-7) 

dt e dx 

where the negative sign appears because the net flow of current dJ was defined 
as being out of the volume element. 

Equation (4-7) takes into account only drift and diffusion. When the 
generation and recombination processes are also included, 

dt r p e dx 

Substituting the values of the drift and diffusion currents from Eqs. (4-3), 
and (4-5) into Eq. (4-6), and differentiating with respect to x, Eq. (4-8) can 
be rewritten as 

ty Po ~ P d d 2 p 

Tt = —^ ~ ^dx (p ^ + D *W (4 " 9) 

This is the continuity equation for positive holes. 

The continuity equation for electrons may be derived by reconsidering 
the three terms separately. From Eq. (1-32), the rate of change of electron 
charge density is related to the generation and recombination rate by 

f*? = H9SU1 (4-10) 

The drift of charge in an electric field $ is given by 

J» r = nefjL n <? (4-11) 

where the superscript — indicates electron flow. The diffusion of electrons in 
the x direction can be represented by a similar equation to Eq. (4-4), i.e., 

where TV is now the number of electrons diffusing across unit area per second, 
and D n is the diffusion constant for electrons. Since electrons have a negative 
charge, the current density carried by this flow of electrons per second per 
unit area is 

7 D - i= - e N=eD n ^ (4-12) 

When the drift and diffusion current densities are considered, we see that a 
net conventional current flow to the right is caused by electron motion to the 



94 THE CONTINUITY EQUATION CHAP. 4 

left. Thus the corresponding equation to Eq. (4-7) is 

8n = 1 dj 

dt e dx 

and the continuity equation for electrons becomes 

8n n n - n d ' ^ d 2 n ,a-i*\ 

+ fi n — (n#) + D n .5-3 (4-13) 



dt r„ nn dx K ' n dx 



4-2 Application of the Continuity Equations 

to the Abrupt p-n Junction Operating with a Constant Current 

The continuity equations may be solved under various conditions of operation 
ofthe/?-« junction. A simple, yet most important, case is that of a constant 
current flowing across a junction, since the solution of this problem leads to an 
understanding of the junction transistor. When the total junction current 
density is a constant, the hole and electron densities in the two regions are 
also constant with respect to time. Thus, wherever the continuity equations 
are applied, we can put 

d JL = — = 
dt dt 

The continuity equations for holes and electrons can now be written as 

= ^ + ,„« + ^S (4-15) 

In Sec. 3-5 there was a discussion of the four current densities which can 
flow across the junction. When the junction is reverse biased so that — eV 
> 4kT, saturation currents J% and / 2 + are flowing as shown in Fig. 4-2(a). 
Both currents flow from regions where the respective charges are minority 
carriers and enter regions where they add to the majority carrier density. 
The total reverse saturation current is small so the majority carrier densities 
are hardly changed by the incoming charges, but the regions which supply the 
charge are greatly affected by the loss of their minority carriers. Close to the 
depletion regions on both sides of the junction, the minority carrier densities 
drop because the saturation currents are removing minority charges. There- 
fore, the continuity equations will be solved only for the minority charges 
close to the junction as these are the cases of maximum interest. 

When the junction is forward biased with eV > 4kT, the two saturation 
currents can be neglected in comparison with the forward currents Jl and 
/?. As shown in Fig. 4-2(b), these forward currents transfer charges from 
the majority carrier regions to the minority carrier regions. The injected 



SEC. 4-2 



APPLICATION OF THE CONTINUITY EQUATIONS 



95 





p-type ! Depletion 
j regions 

i 


n-type 




Electron current flow \ 
from minority \ 
carrier source -^ \^ 




i 


Fermi level 1 %^ 
Reverse bias 













Fermi level 




Distance - 
(a) 


Hole current flow 
from minority 
current source 







p-type 



Depletion 
regions 



n-type 



Minority carrier 
injection (electrons) 



pq Forward bias 

S Fermi level 




Fermi level 



J\ + ""**■ Minority carrier 
injection (holes) 



FIG. 4-2. Diagram showing the carriers flowing across a p-n 
junction (a) under reverse bias, (b) with forward bias. Arrows 
indicate carrier flow direction. 



96 THE CONTINUITY EQUATION CHAP. 4 

minority carrier densities will alter the thermal minority carrier equilibrium 
close to the junction, and so the solution of the continuity equations for the 
minority carrier regions will again be important. 

In the depletion regions, there are very few mobile charge carriers present, 
and so little recombination takes place. We can therefore assume that hole 
and electron currents are, separately, continuous across the depletion region. 
This means that it is necessary to solve the continuity equations only for the 
minority carriers on the two sides of the junction to gain a knowledge of the 
total junction current. In general, we will solve the hole continuity equation 
to find the current carried by holes at the edge of the n-typt region, and we 
will solve the electron continuity equation to find the current carried by 
electrons at the edge of the/?-type region. Assuming no recombination in the 
depletion regions, the sum of these two currents is the constant total current 
throughout the material and across the junction. 

Care must be taken before such restricted conditions are applied to a 
practical case. When very small currents are flowing, recombination may be 
a significant factor. This is particularly noticeable in silicon where the intrinsic 
carrier densities are much lower than in germanium at the same temperature. 
In addition, when the continuity equations are solved only for the minority 
carrier regions, it is on the assumption that the majority carrier levels remain 
at their thermal equilibrium values even though a current is flowing. We will 
distinguish among three conditions of operation for p-n junctions. Very small 
current operation refers to the case where the junction is reverse biased or 
operated in the forward direction with a very small current so that recombina- 
tion in the depletion region must be taken into account. Small-current 
operation occurs when the current density is higher than the previous case 
but not so large that the majority carrier density is disturbed. This applies to 
a typical junction operated with a forward current of a few milliamps. 
Under large-current operation of a p-n junction, such a high current density 
is flowing that the majority carrier density is affected and both continuity 
equations must be solved in both regions to get an accurate analysis of carrier 
density distributions. In most of this book, small-current operation will be 
assumed. However, all three cases are considered in Sec. 9-7 where the 
variation of alpha with emitter current is discussed. 

Close to a typical p-n junction, it is found that the diffusion current density 
is much larger than the drift current density under the condition of small- 
current operation. This is because the minority carrier levels on both sides 
of the junction are substantially modified by the flow of current whether the 
junction is forward or reverse biased as shown in Fig. 4-2. In the following 
analysis in this chapter, we shall neglect the drift terms in the continuity 
equations by comparison with the diffusion terms. (The justification for this 
approximation can be seen by inserting typical values for all the quantities 
involved in the continuity equations.) Equations (4-14) and (4-15) then 
become 



SEC. 4-2 



APPLICATION OF THE CONTINUITY EQUATIONS 



97 



Po 



and 



U "dx 2 



n - n d 2 n 



(4-16) 
(4-17) 



The solution of these two equations for a p-n junction can be found using 
the same coordinate system for the two equations and putting x = at the 
metallurgical junction. This is the most logical method but great simplifica- 
tion can be achieved by using two different coordinate systems as shown in 
Fig. 4-3. For the ft-type material, x is measured in the positive direction to the 
right, and x = coincides with the edge of the «-type depletion region. In 
the p-typQ material, the x coordinate is similarly measured from the edge of 



Metallurgical 
junction 



p-type 
(Hole density iV a ) 



Thermal equilibrium 
hole density 



Electron density 

N at edge of 
p-type depletion 
region .. 



/ 



^T 



Depletion 
regions 



n-type 
(Electron density N d ) 



— Hole density 
P at edge of 
n-type depletion 
region 



Thermal equilibrium 
hole density 

PnO 



Electron continuity 

equation (4-13) 
solved for this region 



Direction x 



x=0 x=0 

for for 

Eqs. (4-13) Eqs. (4-9) 

and (4-17) and (4-16) 



Positive hole continuity 

equation (4-9) 

solved for this region 



Direction x — *■ 



FIG. 4-3. Diagram showing the two coordinate systems used to 
simplify the solution of the two minority carrier continuity 
equations. 

the/7-type depletion region, but, here, the positive direction of x is to the left. 
The two equations are solved separately for the two regions and the situation 
will be clear if it is remembered that in each case, x = signifies the edge of 
the depletion region and x is positive when moving away from the junction 
into the bulk material. 

Consider first the solution of Eq. (4-16) in the rt-type material. We can 
write Eq. (4-16) in the form, 

d 2 Pn Pn ~ PnO (A\K\ 

where the additional subscript shows that the equation is restricted to the 



98 THE CONTINUITY EQUATION CHAP. 4 

rt-type material. Since D p r p can be considered constant, the solution is of the 
form 

where K x and K 2 are constants of integration. This equation applies to the 
tt-type material shown to the right-hand side of Fig. 4-3. 

At a large distance from the junction, the positive hole density must 
approach its equilibrium value in the bulk material. Hence, as 

X-^GO, Pn->PnO 

and so 

°^ exp (vfe) 

giving K 2 = 

The solution of Eq. (4-18) is, therefore 

p„-, n0 = * 1 exp(vi=rj 

From Fig. 4-3, we can write p n = P at the edge of the depletion region where 
x = 0. Hence, 

P-P n0 = K, 

and 

Pn ~ PnO = (P ~ Pno) ™V ( /= ) (4-19) 

Equation (4-19) applies to both forward and reverse-biased junctions by 
inserting an appropriate value of P, as will be done later. For a change in x 
of value Vd p t p , the charge density changes by a factor 1/exp because of 
recombination. The distance, 

L p = VWp (4-20) 

is known as the diffusion length for positive holes in this region, and it can be 
calculated from the values of the diffusion constant and the lifetime. Re- 
writing Eq. (4-19), 

Pn ~ PnO = (P~ Pno) ™ P (^) (4-21) 

in the «-type region. The application of this equation to junctions operating 
under reverse and forward bias is given in Sees. 4-4 and 4-5. 

The continuity equation relating to electrons in the p-type material can be 
solved in a similar manner. From Eq. (4-17), we can write 

" n p _ n p ~ H P0 (A Q'Vt 



SEC. 4-2 



APPLICATION OF THE CONTINUITY EQUATIONS 



99 



where the subscript p restricts the equation to the /?-type material. The 
diffusion length for electrons in this region is L n , where 



L n = VD n r n 

and writing N for the electron density at the edge of the depletion region of 
the /7-type material, 



! p0 



(N - n p0 ) exp 



fei 



(4-23) 



by comparison with the argument in the previous paragraph. Equations 
(4-21) and (4-23) apply to reverse or forward-biased conditions of an abrupt 
p-n junction. 

4-3 Calculation of the Charge Densities 
at the Edge of the Depletion Regions 

We will have completed the solutions of the continuity equations when P and 
N are determined. Under conditions of small-current operation, the majority 



p-type 



n-type 



Electron density 

, n in bulk material ™ 



E„ = E n + eV 7 



Donor density 



eV T = e(V B -V) 
Fermi level 



Acceptor density 
N„ 



N, 



T^T^- , Fermi level 
t Forward 

bias eV 



P Hole density p n0 

in bulk material 



Distance — *• 

FIG. 4-4. Energy band diagram of a forward-biased p-n junction 
showing energy levels and carrier densities in the two regions. 



carrier densities on both sides of the junctions are assumed to be undisturbed 
by the current flow. The minority current densities close to the depletion 
regions are determined by the donor and acceptor densities, and the Boltz- 
mann factor as shown below. 

The Boltzmann factor [Eq. (2-14)], 



F(E) = exp 



(V) 



(4-24) 



100 THE CONTINUITY EQUATION CHAP. 4 

is the probability, expressed in terms of the Fermi level, that an electron state 
at an energy E is occupied. Writing E n as the lowest energy level in the con- 
duction band of the /t-type material and E p as the lowest energy in the con- 
duction band of the p-type material, and by reference to Fig. 4-4 and Eq. 
(3-19), 

V T = V B - V (4-25) 

where V is the applied voltage across the junction, and 

E p = E n + eV T (4-26) 

If the applied voltage, V, were zero, the Fermi levels on the two sides of 
the junction would be in line. This means that the probabilities of occupation 
of states at a given energy on the two sides of the junction would be equal. 
In Fig. 4-4, however, a forward bias is shown. In this case, the Fermi level 
of the p-type side is eV below the Fermi level of the n-typc side. For a given 
energy, E, in the conduction band, the ratio of the probability of occupancy 
in the «-type to the p-typQ is 



exp [(Ep - E)/kT] 
exp [(E F - eV - E)/kT] eXp 



© 



where E F is the Fermi level on the «-type side of the junction. 

A current of electrons flows in the conduction band across the junction to 
equalize the probability of occupancy across the two sides. Since we have 
assumed small-current operation, this flow of charge does not disturb the 
majority carrier density in the n-type region but raises the minority carrier 
density in the/?-type material by the factor exp (eV/kT). A small, continuous 
current is necessary to sustain this new minority carrier density on the left- 
hand side of the junction (p-type region) because diffusion and recombination 
are removing charge into the bulk material. Since the minority carrier density 
under thermal equilibrium in the bulk p-type material is n p0 , the increased 
electron density close to the depletion region is 

N = n p0 exp (g) (4-28) 

A similar argument can be applied to the valence band. The probability 
for the occurrence of holes has previously been given as 

1 - F(E) a exp {^~) (4-29) 

from Eq. (2-16), where E is now an energy lying within the limits of the 
valence band. The ratio of the probabilities of finding holes at an energy Em 
the valence band of the p-type and the «-type material is 



exp [(£ - E F )/kT) 
exp [(E - E F - eV)/kT] 



= -p (9 (4 - 3o) 



SEC. 4-3 CHARGE DENSITIES AT THE EDGE OF THE DEPLETION REGIONS 101 

as before. In this band, therefore, a current of positive holes moves from left 
to right to raise the minority carrier (hole) density at the edge of the «-type 
material by the factor exp (eV/kT). Since the hole density in the bulk n-type 
material is p n0 , 

P = Pn0 exp{j^ (4-31) 

and the continuity equations can be written as 

Pn ~ Pno = Pno [exp \j^j - 1 1 exp \-j^J (4-32) 

h(SH exp (if) (4 - 33) 



n n - n 



P o — "pO 



Care must be taken to apply these equations only to the regions where they 
were derived and only under small-current operation. Equations (4-32) and 
(4-33) apply when the junction is either reverse or forward biased since 
Eqs. (4-28) and (4-31) hold for negative or positive values of V, although we 
have described here the flow of current in the positive case only. 

4-4 Solution of the Continuity Equations 

for a Reverse-biased Abrupt p-n Junction 

The continuity equation solutions, Eqs. (4-32) and (4-33) apply directly to 
this case. Since the junction is reverse biased, Kis negative, and exp (eV/kT) 
< 1. Let us first examine the density of holes in the «-type region. 
From Eq. (4-32), at the edge of the depletion region, x — 0, and so 

Pn = P = Pno exp ( ~ J < p n0 (4-34) 

Thus the minority charge density at the edge of the depletion region is less 
than the density in the bulk material. The physical reason for this is that the 
current J 2 + is carrying holes across the junction into the p-type material, and 
these holes are being supplied from the bulk n-type material by diffusion. 
For electrons in the p-type material, at x = 

n p = N = n p0 exp (^,1 < n p0 (4-35) 

by a similar argument. In this case, J 2 is the electron current which flows 
across the junction and depletes the minority carrier level in the p-type 
material. 

When the junction is biased in the reverse direction with a voltage greater 
than about 0. 1 volt at 300° K, we can write 



-eV > AkT thus exp W=\ « 1 



102 THE CONTINUITY EQUATION CHAP. 4 

This corresponds to the case when J £ and J£ carry almost all the current 
across the junction. From Eq. (4-5), 

J m - eD p dx 

where J^ { is defined in the positive x direction, i.e., to the right in the «-type 
material. Hence, 

J ™ = ~^ p ^ eXp (l~) (4 " 36) 

by differentiating Eq. (4-32). The negative sign in Eq. (4-36) indicates that 
the diffusion current flow is to the left in the tf-type material. 
For electrons in the /?-type material, from Eq. (4-12), 

/i-.A.J-«A.te oq ,(^) (4-37) 

by differentiating Eq. (4-33). This current is defined to be in the positive x 
direction in the /?-type material, i.e., from right to left. 

At the edge of the «-type depletion region, x = 0, and the hole current to 
the left as defined in Sec. 3-5 is 

Ji=eD p P -f (4-38) 

At the edge of the /?-type depletion region, x = 0, and the electron current to 
the left is defined as 



71 L 



J 2 -=eD n '^ (4-39) 



From Sec. 3-5, the total current density defined in the forward direction is 

since we have assumed J$ = /£" = for a junction with a large reverse bias. 
Hence, 



eDp P^ + eD 



ng) (4-40) 



Thus a reverse-biased junction will pass a current in the reverse direction given 
by Eq. (4-40). Since p n0 and n p0 are small compared to N d and N a , the reverse- 
biased junction current density is small. 

Figure 4-5 shows the result of a computation of Eqs. (4-32), (4-33), 
(4-36), and (4-37) for the case of the p-n junction used as an example in 
Chap. 3. The minority carrier densities, which are shown in the upper set of 
curves, are small close to the depletion regions and increase exponentially 
with density in the bulk material until the thermal equilibrium value is 
reached. The two diffusion currents are plotted as full lines in the lower set 
of curves in Fig. 4-5. The diffusion current of positive holes in the w-type 



SEC. 4-4 CONTINUITY EQUATIONS FOR A REVERSE-BIASED ABRUPT p-n JUNCTION 



03 



p-type 
1 ohm cm 



n p0 = 1.65xlO u cra'\ 



Electron 
density 




Hole 
density 



10 7 



Jl 



n-type 
0.1 ohm cm 



p =3.00xl0 10 cm -3 




0.02 0.04 0.06 0.08 0.10 



Distance from edge of 
depletion region cm 



Distance from edge of 
depletion region cm 



(a) 



p-type 
1 ohm cm 



Total current density J 



Current 

density 

amps cm" 




Holes ^ 



Electrons 
(minority carriers 



0.10 0.08 0.06 0.04 0.02 



rc-type 
0.1 ohm cm 



Total current density J 
Electrons (majority carriers) 



Holes (minority carriers) 



0.02 0.04 0.06 0.08 0.10 



Distance from edge of 
depletion region cm 



Distance from edge of 
depletion region cm 



(b) 



FIG. 4-5. Minority carrier and current density for a reverse- 
biased abrupt p-n junction. 



104 THE CONTINUITY EQUATION CHAP. 4 

material is labeled "holes (minority carriers)" and is traveling towards the 
left-hand side of the diagram and constitutes a current in this direction. The 
diffusion flow of electrons in the p-type material is labeled " electrons 
(minority carriers)" and is moving to the right. Since electrons carry a 
negative charge, the current associated with the electron motion is to the left. 
In both materials, the current which is carried by the minority carriers is at 
its maximum at the edge of the depletion regions. Further into the bulk 
material, recombination takes place, and the current is mainly carried by 
majority carriers. 

Values of the constants that have been used to draw Fig. 4-5 are given 
below. The voltage across the junction was —0.15 volt. 

p-type germanium: 

P P = 1 ohm cm N a = 3.8 x 10 15 cm" 3 

fi n = 3320 cm 2 volt" 1 sec" 1 n p0 = 1.65 x 10 11 cm" 3 

D n = 86 cm 2 sec" 1 L n = 0.05 cm 

«-type germanium: 

Pn = 0.1 ohm cm N d = 2.1 x 10 16 cm" 3 

fji p = 1200 cm 2 volt" 1 sec" 1 p n0 = 3.0 x 10 10 cm" 3 

D p = 31 cm 2 sec -1 L p = 0.05 cm 

The total current density across the junction is 



J = 



eD p ^p + eD n ^f\ = -4.84 x 10" 5 amps cm 
L p L n J 



This value ignores the effect of surface leakage which is present in a practical 
case. 



4-5 Solution of the Continuity Equations 

for a Forward-biased Abrupt p-n Junction 

The solutions of the continuity equations are limited to the small-current 
condition. For positive holes in the «-type material, Eq. (4-32) still applies. 

Pn - Pno = />no[exp \j^j - lj exp \j^J (4-32) 

where Kis now positive and exp (eV/kT) > 1. Similarly, for electrons in the 
/7-type material, Eq. (4-33) is still valid. 

n v - n p0 = n p0 j^exp \j^j - lj exp \^j (4-33) 

where V is positive. 



SEC. 4-5 CONTINUITY EQUATIONS FOR A FORWARD-BIASED ABRUPT p-n JUNCTION 105 

The diffusion current of holes at any point in the n-type material is, 

J » - ~ eD " i = e jr,'" [ exp (w) ~ l ] exp (~rj 

= e^(p n - Pn0 ) (4-41) 

defined in the positive x direction in the «-type material, i.e., from left to 
right. Similarly, 

J* = eD n g = - e ^ (n p - n pB ) (4-42) 

defined in the positive x direction in the />-type material, i.e., from right to 
left. 

When the junction is forward biased with a voltage greater than about 
0.1 volt at 300° K, 



eV > AkT and exp h— » 1 



m 



Thus the reverse saturation currents (J '% and J 2 ) are negligible, and the total 
current across the junction can be written as 

J = J1 + Jl 

where 



T + D P t eV \ 



when x = in the tf-type region and /f is defined to flow from left to right. 
Also, 



J x = ,_, p0 exp^j 



when x = in the /?-type region and J 1 is defined to flow from left to right. 
Taking account of the directions of current flow, therefore, 

H e ?> + e r>) exp (S) ^ 3 > 

The curves of charge density and current for the typical p-n junction are 
plotted in Fig. 4-6. The applied voltage is 0.15 volt in the forward direction. 
On both sides of the depletion region, the minority carrier density is increased 
because of the injection of minority charges from the other side of the junc- 
tion. As the minority carriers diffuse away from the junction, recombination 
occurs and the minority carrier current density drops. For instance, in the 
«-type region on the right-hand side, the minority carriers (holes) recombine, 
and the current becomes electronic. At a distance L n from the edge of the 
depletion region only 1/exp of the initial hole diffusion current is still being 
carried by holes. 



p-type 
1 ohm cm 




^ — 


] 


u a7 
n p0 = 1.65xlO n cm" 3/ 


- 
- 


Electron 


] 


density 


- 


cm -3 






] 





n-type 
0.1 ohm cm 




10 13 - 
10 12 - 






10 11 - 






10 10 - 


Vp n0 =3.00xl0 10 cm" 3 




10 9 - 






10 8 - 


Hole 

density 

cm -3 




10 7 - 







0.10 0.08 0.06 0.04 0.02 



0.02 0.04 0.06 0.08 0.10 



Distance from edge of 
depletion region cm 



(a) 



Distance from edge of 
depletion region cm 



p-type 
1 ohm cm 

Total current density J 



Holes 
(majority carriers)^S 




Injected electrons 
(minority carriers) 



Current 

density 

amps cm -2 

- 1.6X10- 2 

- 1.4 xKT 2 

- 1.2xl0- 2 
lx lO -2 

8xl0 -3 
6xl0 -3 - 
4x10" 
2x10" 



n-type 
0.1 ohm cm 



Total current density J 
Electrons (majority carriers) 




0.10 0.08 0.06 0.04 0.02 

Distance from edge of 
depletion region cm 



Injected holes (minority carriers) 



0.02 0.04 0.06 0.08 0.10 



(b) 



Distance from edge of 
depletion region cm 



FIG. 4-6. Minority carrier and current density for a forward- 
biased abrupt p-n junction. 



SEC. 4-5 CONTINUITY EQUATIONS FOR A FORWARD-BIASED ABRUPT p-n JUNCTION 107 

In this example, the applied voltage is assumed to be constant and so dpjdt 
and dn/dt are everywhere zero. Assuming isothermal and isotropic conditions 
throughout, the sum of the minority and majority currents is constant across 
all regions, and this has been indicated on the diagram. In the depletion 
regions, there is a scarcity of mobile charges, and recombination is very 
small. The hole and electron currents are assumed to be unchanged while in 
these regions. 

4-6 A Brief Discussion of the p-n Junction 

In the foregoing, the continuity equations have been solved for the reverse 
and forward bias conditions. Both cases are important in understanding the 
junction transistor and will be referred to in Chap. 5. At this stage, it is well 
to summarize briefly the physical meaning of the solutions. 

When an abrupt p-n junction is reverse biased, the minority carrier 
densities at the edge of the two depletion regions are very low. The total 
reverse junction current density is also small, and it is determined by the 
minority carrier densities in the bulk material [Eq. (4-40)]. 

When the junction is forward biased, the majority of the current is carried 
by injected minority carriers. Holes will move from the /?-type side to the 
«-type material where they become minority carriers. Electrons travel from 
the «-type material across the junction and become minority carriers in the 
/?-type side. These injected currents increase the minority carrier densities 
close to the depletion regions, and so a diffusion of minority carriers away 
from the junction takes place (Fig. 4-6). As these minority carriers move 
away from the depletion region, recombination occurs. In the p-type material, 
the injected electrons gradually recombine with the positive holes, and the 
proportion of the current carried by the majority carriers (holes) increases 
with distance from the edge of the depletion region. At a distance 4L n in the 
/7-type region most of the recombination has already occurred and nearly all 
the current is carried by holes. In the «-type material, recombination also 
occurs with the result that the transfer of current from holes to electrons is 
nearly complete at a distance 4L P from the edge of the «-type depletion region. 

At this point, we can distinguish between the p-n junction and an ohmic 
contact. The sole purpose of an ohmic contact is to form a connection be- 
tween the semiconductor region and the external lead so that current can be 
passed into, or out of, the device. Unlike the p-n junction, therefore, the 
ohmic contact should have a linear voltage versus current characteristic, and 
it should behave as a small ohmic resistance. Ohmic contacts are frequently 
made between a metal and the semiconductor region. The surface structure of 
the semiconductor is often deliberately damaged in some way around the 
contact. This increases the recombination rate in the vicinity of the contact, 
and, by keeping the carrier concentration near equilibrium, prevents to some 
extent the formation of a rectifying junction. Ohmic contacts may be formed 
by soldering, alloying, or other types of bonding process. 



108 THE CONTINUITY EQUATION CHAP. 4 

4-7 The Incremental Resistance of a Forward-biased p-n Junction 

The total current density, J, in a forward-biased junction is given by Eq. 
(4-43) to be 

'-(^ + .4.g)«p(g) (4-43) 

For a junction of area A, the total current is 

I = JA (4-44) 

An incremental resistance for the junction can be defined to be 

J_ 
dV 1 dV A 



dl A dJ _£_ 
kT 

kT 
el 



{ e ^ + e t np °) Qxp {S) 



(4-45) 



At room temperature, T = 300° K and kT/e = 0.0258 volts. Hence, 

0.0258 



ohms (4-46) 

when / is specified in amps. r e is the resistance presented to small changes in 
the voltage across the junction. For a given junction current, the incremental 
resistance, from Eq. (4-46), is independent of the area of the junction. 



4-8 Einstein Equation 

There is an important relationship, known as the Einstein equation, which 
exists between the mobility and diffusion constant. It will be derived here by 
reference to the current flow in a p-n junction. 

In the conduction band of Fig. 4-7, there are two current density com- 
ponents flowing across the junction, /f and J^. J 2 is the current which 
flows down the potential hill from the /?-type to the «-type region. It is, 
therefore, a drift current of electrons of the form, 

J 2 = ne^ (4-47) 

where $ is the electric field, and n is the density of electrons in the depletion 
regions. The current, Jl, is the result of diffusion of electrons from the 
tf-type region, where electrons are majority carriers, to the /?-type material, 
where they are in the minority. Thus, 

r- r» dn 

A = eD "Tx 



SEC. 4-8 



EINSTEIN EQUATION 



109 



We know that when the applied voltage is zero, 

dn 



or 



ne\L n £ = eD n 



dx 



Integrating across the junction as shown in Fig. 4-7, 

A, r N *dn 






or V B = — log e -— = — 

from Eq. (3-18). Thus, 



eV t 



kT 






e 
kT 



p-type 



n-type 



Electron 
density 



c 

w 



eVn 




Electron 
density 



(4-48) 



a b 

Distance — ► 

FIG. 4-7. p-n junction with no applied voltage showing electron 
densities and currents. 

From a consideration of holes in the valence band, we can show that 



ts. = — 

D p kT 



(4-49) 



This relationship between the mobility and the diffusion constant is the 
Einstein equation. 



BIBLIOGRAPHY 

Middlebrook, R. D., An Introduction to Junction Transistor Theory, New 
York: John Wiley & Sons, Inc., 1957 



110 THE CONTINUITY EQUATION CHAP. 4 

Shive, J. N., Properties, Physics and Design of Semiconductor Devices, New 
York: D. Van Nostrand Company, Inc., 1959 

Shockley, William, Electrons and Holes in Semiconductors, New York: D. 
Van Nostrand Company, Inc., 1950 

Spangenburg, Karl R., Fundamentals of Electron Devices, New York: 
McGraw-Hill Book Company, Inc., 1957 

PROBLEMS 

A- 1 In the derivation of the continuity equation, the effects of only three 
phenomena were taken into account. If the electric field in a semi- 
conductor is related to the temperature gradient by the equation, 

dS = KdT 

where $ is the electric field, T is the temperature, and K is a constant, 
derive the term which must be added to the continuity equation to 
account for this phenomenon. 

4-2 If the electric field in a semiconductor is zero and the injected electron 
concentration varies with an angular frequency oj, i.e., 

n(x, t) = n(x) exp (jot) 

show that the continuity equation becomes 



<Pn 
dx 2 



- HP)- 



4-3 Discuss the continuity equation for electrons, identifying the terms that 
account for the three phenomena affecting the charge distributions and 
current flow in a semiconductor. How are these terms derived ? What 
is the form of the equation if no field is applied and a steady state has 
been reached ? 

4-4 It has been stated that a very small proportion of the minority carrier 
current close to a semiconductor p-n junction is carried by drift of the 
minority carriers in the electric field. Using the values for the p-n 
junction given in Sec. 4-4, compare the relative magnitudes of the 
minority drift and diffusion currents at a distance of two diffusion 
lengths away from the edge of the depletion region in the «-type 
material. 

4-5 A germanium p-n junction consists of two regions 2 mm long and 
having an area of 1 sq mm. The resistivity of the n-typQ region is 3 ohm 
cm. If the lifetime of holes in the n-type region is 10 microseconds and 
the /7-type region is heavily doped compared to the n-typc, calculate 
the approximate saturation current at 300° K. 



PROBLEMS I I I 

4-6 Explain the phenomena of diffusion, drift and recombination and 
show how they are related in the continuity equation. 

4-7 Using the data given in Sec. 4-4, determine the value of x in the fl-type 
material at which 95 per cent of the hole minority carriers have re- 
combined. At what distance from the edge of the depletion region do 
the majority and minority currents contribute equally to the total 
current density? 



4-8 Show that Eq. (4-40) can be expressed as 

j _ J PnpLp n p0 L n \ 
\ J p J n J 



or 

/ = - en\ 



\L p N d + L n N a ) 



4-9 Describe what is meant by "small-current" operation. Show why it is 
not necessary to solve the continuity equation applying to majority 
charges to obtain a solution for the charge density and current flow 
in a p-n junction operating under small-current conditions. 

4-10 Show that the ratio of hole current to electron current across a semi- 
conductor junction can be expressed as 

J_p _ QpLn 

Jn a nL p 

4- 1 1 Find the current flowing across a p-n junction of area 10 " 6 m 2 at 300° K 
when the applied voltage is 2 volts in the reverse direction. The 
resistivities of the p- and n-type regions are both 1 ohm cm. Assume 
L n = 0. 1 cm and L p = 0.04 cm. 

4-12 Summarize and discuss the principal electrical characteristics of a p-n 
junction as given in Chaps. 3 and 4, dealing in particular with (a) 
depletion widths, (b) incremental resistance and capacitance, (c) 
reverse and forward current versus voltage characteristic. 



5 



The 

Junction 

Transistor 



In the preceding chapters the theory of electrical 
conduction in semiconductors has been applied 
only to the rectifying, abrupt p-n junction. In this 
chapter, the same theory is extended to analyze the 
junction transistor. 

Historically, semiconducting rectifiers date back 
to the "crystal and cats-whisker" type of detector 
which was used in the early days of radio. This 
simple device operated as a rectifier when the cats- 
whisker was able to find a spot on the crystal where 
the impurity content was such as to create a p-n 



SEC. 5-1 THE npn TRANSISTOR 113 

junction. The development of the point-contact germanium diode prior to 
1945 was the result of manufacturing a device with more control over the 
point contact and the semiconducting base material. In 1948, Bardeen and 
Brattain announced a point-contact transistor. This was a device having two 
contacts on an n-type germanium base, and experiment showed that such an 
arrangement could provide a current and power gain when operated with 
suitable applied potentials. This type of transistor is rarely used now because 
of manufacturing difficulties and the presence of a high noise level. The 
theory of the point-contact transistor is not completely understood at the 
present time. 

Developments in the semiconductor field were rapid after 1945, and, by 
1949, the theory of p-n junctions had been established. At this time, Shockley 
announced, and gave the theory of, a junction transistor, and, within a few 
years, this new device was commercially available. The junction transistor is 
composed of an npn or pnp sandwich where the junctions may be abrupt or 
graded. In this chapter, only transistors using abrupt junctions will be 
considered and the main characteristics of these devices will be discussed. 



5-1 The npn Transistor 

In the particular p-n junction which was used as an example in Chaps. 3 and 4, 
the conductivity of the/?-type material was 1 mho cm -1 and that of the n-type 
material was 10 mho cm -1 . When this junction is forward biased with an 
applied voltage of 0.15 volts, the electronic current density across the junction 
is 

^^-°exp(^ = 1.56 x 10- 2 ampcm- 2 
The positive hole current density across the junction is 

^^° ex p(j^) = L02 x 10- 3 ampcm- 2 

The electron current therefore carries more than 90 per cent of the charge 
flow across the junction. If the conductivity of the n-type material is in- 
creased to, say, 100 mho cm -1 while the p-type conductivity is reduced to 
0.1 mho cm -1 nearly all of the current passing across the forward-biased 
junction will be carried by electrons. The energy band diagram of this junction 
is shown in Fig. 5-1. 

When the junction is forward biased with a voltage greater than 4kT/e, the 
two reverse saturation currents can be neglected, and, if we also neglect the 
small positive hole current in the forward direction, the total current is simply 

J = Jl (5-1) 



14 



THE JUNCTION TRANSISTOR 



CHAP. 5 





100 mho cm l 0.1 mho cm -1 




/ 


1 


Conduction band 




electrons "^s 

Fermi level 


a 


„ ,, J Fermi level 
Forward bias 




^/""~ 




Valence band 




^/^ 




Distance — *- 



FIG. 5-1. Energy band diagram of a p-n junction where the 
conductivity of the /z-region is much higher than that of the 
/j-region. 

The electron current J I is shown in the diagram to flow from the «-type to 
the /?-type region, Thus, the minority carrier density close to the junction in 
the /?-type material has been increased because of electron injection. 

Next consider Fig. 5-2 which represents an npn transistor. A second 
junction has been made by adding a piece of «-type material to the right-hand 
side of the />-type semiconductor. The width of the central /?-type material in 
the sandwich has been made very small, and the second junction is reverse 




Electron flow 
across junction 



Fermi level 



Reverse saturation 
current of electrons 



Reverse bias 

Fermi level 



Emitter 



Base 
Distance 



Collector 



FIG. 5-2. Energy band diagram of an npn transistor. 



SEC. 5-1 



THE npn TRANSISTOR 



115 



biased. (The methods of manufacturing transistors are considered in Sec. 
5-6.) If the width of the central region is made much smaller than the electron 
diffusion length, L n , little recombination occurs before the electrons diffuse 
across to the second junction. 

In Sec. 4-4, it was shown that the current across a reverse-biased junction 
is dependent upon the minority carrier density in the bulk material. In the 
/7-type central region, the minority carrier density has been increased because 
of the injection of electrons from the left-hand junction. The reverse satura- 
tion current of the right-hand junction is thus increased, and substantially all the 
injected current flows through the central region and across the second junction. 

This device is an npn junction transistor and the three regions are identified 
in Fig. 5-2 as the emitter, the base, and the collector. Under normal operating 
conditions, the emitter-to-base junction is forward biased and the base-to- 
collector junction is reverse biased. 

Emitter current — ► Collector current — *■ 

Direction of electron flow Direction of electron flow 

npn 

Emitter Collector 



Current 
limiting 
resistance 

Emitter-to- 
base forward 
voltage battery 
2 volts 



<S> 



\ 



Direction of 
conventional 
current flow 



/ 



<& 



Base 



Collector-to-base 
t_L reverse voltage 
"y" battery 

2 to 40 volts 



FIG. 5-3. The npn transistor: symbol and connections. 

The flow of current across the emitter-to-base junction is known as the 
emitter current, and this current is mainly controlled by the emitter-to-base 
voltage. Nearly all of this current flows across the base into the collector 
region, and, provided the collector-to-base junction is reverse biased, the 
collector current is almost independent of the reverse applied voltage (see 
Fig. 5-10). 

The symbol which is widely used to represent the transistor is given in 
Fig. 5-3 where it is seen to consist of two leads connected to a base region. 
This symbol was first used for the point-contact transistor and has been 
retained for the junction type. The emitter lead is identified by having an 
arrow on it ; this arrow shows the direction of flow of the conventional current. 
For the npn transistor of Fig. 5-3, the electrons flow into the emitter lead, and 
so the direction of conventional current is out from the base. 

It is not yet apparent that the transistor can be used as an amplifier. The 
amplifying properties of a transistor are discussed briefly in Sees. 5-4 and 5-5 
and in some detail in Chaps. 6, 7, and 8. At this point, some general remarks 
on the operation of the transistor are in order. 

It has been shown that the emitter current flows through the base and into 
the collector region, and it is found that the collector current is almost 



116 THE JUNCTION TRANSISTOR CHAP. 5 

independent of the collector-to-base voltage provided this junction is reverse 
biased. Therefore, by varying the emitter current, we will vary the collector 
current as well, and so control the power which will be dissipated in a load 
resistance placed in the collector circuit. When the collector voltage is higher 
than the emitter supply, the power dissipation in the collector circuit is 
controlled with the expenditure of a smaller power in the emitter circuit, and 
we have achieved a form of amplification. This is not the only, nor even the 
most usual, method of operating a transistor; it does indicate, however, that 
the device has considerable potentialities as an amplifier. 

5-2 The pnp Transistor 

Figure 5-4 shows the energy band diagrams of a pnp transistor. The emitter is 
made of high conductivity p-type semiconductor, and the base region is of 
low conductivity n-type material. Thus the emitter current which flows when 
the first junction is forward biased consists largely of positive holes which 







r^ 






v / 




/ 


t 


1 Fermi level 


* 


Fermi level 

Positive hole flow 
across junction 


\ h 

^ ' Reverse bias 




i 1 //*" Reverse saturation 

. — «^. / / current of holes 

*" Forward / / 
— hi as // 




Hole flow / 
by diffusion/ 




Emitter 


\__7 




Base Collector 
Distance — *■ 



FIG. 5-4. Energy band diagrams of a pnp transistor. 

travel through the thin base region until they reach the reverse-biased 
collector-to-base junction. Here the holes are collected by the final p-type 
collector region. It will be recalled that, since holes carry a positive sign, the 
upward direction in Fig. 5-4 represents a "downhill" or accelerating potential 
for holes. 

The symbol used for the pnp transistor has the arrow on the emitter lead 
pointing into the base and indicating the direction of flow of conventional 



SEC. 5-2 THE pnp TRANSISTOR I 17 

current. The circuit of Fig. 5-5 is similar to that for the npn transistor except 
for the polarity of the batteries. The pnp transistors are used more frequently 
than npn because they are easier to manufacture (see Sec. 5-6). 

Emitter current — ► Collector current — ► 

Direction of hole flow Direction of hole flow 

pnp 



Current 
limiting 
resistance 

Emitter- to - 
base forward "t=" 
voltage battery 
2 volts 



©Emitter Collector f7\ 
\7 \*ss 



Collector-to-base 

reverse voltage 

battery 

2 to 40 volts 



FIG. 5-5. The pnp transistor: symbol and connections. 

5-3 Characteristics of the Current Flow across the Base Region 

A. Current flow by diffusion. This section will summarize the principal 
characteristics which determine the operation of the transistor. In order to 
emphasize the main physical arguments several simplifying assumptions will 
be made. The analysis will be carried out by the application of the continuity 
equation to the base region of a pnp transistor. The analysis of an npn 
transistor follows a similar pattern. 

Transistors are most frequently used as amplifiers of alternating currents 
in the audio and radio frequency range. Direct analysis of the transistor 
under a-c conditions will not be attempted in this section which is mainly 
concerned with direct-current operation of the device. However, the estab- 
lishment of the d-c characteristics is a necessary step towards an under- 
standing of alternating-current operation. 

Consider the positive hole current which is injected into the base region of 
a pnp transistor. The continuity equation for holes in the fl-type region states 
that 

ty Pno ~ P g(g£) , n d 2 P ,c ? x 

ei = —^r ~ ^isr + ^a? (5 ~ 2) 

The following simplifying assumptions will be made in the first instance : 

1. Small-current, steady-state conditions apply with dp/dt = 0. 

2. Recombination in the base region is very small since the base width is 
much less than L v and so the recombination term in Eq.(5-2) can be neglected. 

3. The voltage drop is confined to the two junctions so that there is no 
field in the base region and d{pS)jdx = 0. 

Under these conditions, the continuity equation reduces to 

Z>,g = (5-3) 



118 



THE JUNCTION TRANSISTOR 



CHAP. 5 



and has a solution of the form 



Ax + B 



(5-4) 



From Sees. 4-2 and 4-3, the hole density at the edge of an «-type depletion 
region in the presence of a junction voltage V is 



Pno exp 



(S) 



(5-5) 



where p n0 is the thermal equilibrium hole density in the /7-type base material. 
In the coordinate system suggested by Fig. 5-6, x = is at the right-hand edge 



p-type 



Metallurgical 
junction 



n-type 



Depletion regions 

(small) / e y \ 

-Hole density p=p n0 exp \-~J 



Hole density 
assuming no 
recombination 



p-type 



Metallurgical 
junction 




Hole 
density 
Thermal equilibrium X * 0^ 
hole density p„ >v \ 



Depletion 
regions 



x = 



W 



Emitter 



Collector 



Distance in base region 



FIG. 5-6. Hole density in the base region of a pnp transistor 
assuming no recombination. 

of the base-to-emitter depletion region, and, if the emitter-to-base direct 
voltage is V EB , 



Pno exp 



m 



at x = 



(5-6) 



The edge of the collector-base depletion region is at x = W in Fig. 5-6 
and, if V CB is the voltage across this second junction, we note that for a pnp 
transistor V CB is negative and usually has a magnitude of a few volts. We can 
write 

— eV CB » kT, and so, at x = W, 



Pno exp 



(*)- 



(5-7) 



SEC. 5-3 CHARACTERISTICS OF THE CURRENT FLOW ACROSS THE BASE REGION | 19 

Substituting these two boundary values for p in Eq. (5-4), 

p.-^(l--i)«p(^) (5-8) 

This equation is plotted in Fig. 5-6. The three simplifying assumptions 
which were used at the beginning of this section, therefore, lead to the con- 
clusion that there is a linear reduction in hole density with distance in the base 
region. The current density flowing across the base under these conditions is 
due entirely to diffusion of positive holes. The diffusion current density is 

/ = y + = -^| = ^exp(^) (5-9) 

Since x does not appear in Eq. (5-9), the diffusion current remains constant 
across the base region. 

B. Emitter efficiency. It was shown previously that by unequal doping of the 
p- and tf-type regions of a junction the current flowing may be made to consist 
largely of positive holes. When this p-n junction forms the emitter-to-base 
junction of a pnp transistor, this condition must be re-examined in the light 
of Eq. (5-9). For positive holes moving from emitter to base, the current 
density is inversely proportional to the active base width W. For electrons 
moving from base to emitter, the current is given by Eq. (4-43) as 

A-=^exp(^) (5-10) 

where n p0 is the thermal equilibrium electron density in the emitter. The ratio 
of the hole current to the total current flow across the junction is known as 
the emitter efficiency, y. Since it is only the hole current which takes part in 
the pnp transistor action, y should be very close to unity for efficient operation 
of the transistor. 

Jj 1 1 

Jt D p p n0 L n 

«uM a i-^ (5-11) 

V p PnoL n a e L n 

from Eqs. (4-48), (4-49), and (2-34), where a b and a e are the conductivities of 
the base and emitter regions respectively. This ratio is higher for the transistor 
than it is for a corresponding p-n junction. The reason for this is that under 
small current conditions the minority charges just at the edge of the two 
depletion regions are maintained at values determined by the applied voltages. 
The current flow of holes in the base region of the transistor, however, is 
greatly increased by the presence of the reverse-biased collector-to-base 
junction. 



120 



THE JUNCTION TRANSISTOR 



CHAP. 5 



C. Base width modulation. Under small-current conditions, the hole density 
close to the emitter side of the base region is unaffected by the presence of a 
second junction at the other end of the base if V EB is constant. However, the 
collector-to-base junction is reverse biased and acts as a sink for minority 
carriers in the base region. Equation (5-9) shows that the positive hole 
current across the base is inversely proportional to the active base width, W. 
In practice, the depletion region at the emitter end of the base region can be 
neglected since the forward bias, V EB is small, but the collector-to-base de- 



p-type 



n-type 



p-type 



Metallurgical 
junction 



Metallurgical 
junction 



-Charge density at x = 0, 

jeV EB \ 
e Pn o exp ^— J 




j2= w; p »° exp 



Edge of depletion 
region for V CB1 



Edge of depletion 
region for V CB2 



W 2 W l B 



Emitter 



Base 
Distance in base region 



Collector 



FIG. 5-7. Diagram showing the change in current density pro- 
duced by a change in collector-to-base voltage, V CB , when V EB is 
a constant. 



pletion width may extend a considerable distance into the base region. Writ- 
ing B for the metallurgical width of the base, from Fig. 5-7, the base depletion 
region close to the collector is of extent B-W. From Eq. (3-13), B-W is 
proportional to {V CB )K Hence, W, and also the hole current through the 
base, are also functions of V CB as shown in Fig. 5-7. 

In normal operation of the transistor, the condition V EB is constant is 
unusual, and, more commonly, the direct emitter current I E is held constant. 
Under these conditions, the modulation of base width due to changes in 
collector-to-base voltage has only a second order effect on collector current. 
Small though this effect may be, it is still greater than the corresponding 
change in current in a simple reverse-biased p-n junction, and it is the main 
factor which determines the incremental collector-to-base resistance in a 
junction transistor. Base width modulation is also responsible for a small 



SEC. 5-3 



CHARACTERISTICS OF THE CURRENT FLOW ACROSS THE BASE REGION 



21 



change in the emitter-to-base voltage-current characteristic when the collector 
voltage changes. 

D. Recombination in the base region. If the three assumptions listed at the 
beginning of this section were completely true, all the positive holes entering 
the base region would pass to the collector. In practice, it is found that 1 or 
2 per cent of the current flows out of the transistor by the base lead. The 
largest component of this current is caused by recombination in the base. 



p-type 



n-type 



p-type 



Metallurgical 
junction 



Metallurgical 
junction 



o 



ep„n ex P 



m 




Small amount of 
^SX recombination 
TangentS. affects slope 
atx = 



Tangent ^W 
atx=W ^s 



W 



Emitter 



Collector 



Distance in base region — *• 



FIG. 5-8. Charge density in the base region showing the effect of 
recombination. [Solution of Eq. (5-12).] 



Taking account of this recombination, the continuity equation can be written 
as 

P - PnO 



Uv dx 2 



(5-12) 



and may be solved to give a more correct analysis of the variation of hole 
density in the base region. This equation is not solved until Chap. 9 but 
Fig. 5-8 shows that the resulting charge distribution is slightly concave in 
shape. The hole current entering the base region is proportional to the 
negative slope of the curve at x = from Eq. (5-9). From Fig. 5-8, the slope 
at this point is clearly more negative than the slope at x = W. The difference 
between the current entering the base region from the emitter and that leaving 
through the collector is the small current in the base lead. 
E. Collector-to-base cut-off current. Another small current is flowing into the 



122 



THE JUNCTION TRANSISTOR 



CHAP. 5 



base region which has hitherto been neglected. This is the saturation current 
of the collector-to-base junction which flows because the collector-to-base 
junction is reverse biased. This current is present even when the emitter 
current is zero and is known as the cut-off current. It is generally designated 
either I CBO or I co . For a typical small germanium transistor at room tem- 
perature, I CBO is a few microamps, and, for a silicon unit, it is usually much 
below one microamp. Although of small magnitude, the temperature varia- 
tion of I C Bo is high, and, in some circuits, the cut-off current can markedly 
affect the total collector current (see Chap. 8). 

F. Diffusion capacitance. When the transistor is operated in the usual manner, 
we have shown that the minority charge density decreases linearly with dis- 
tance across the base. This charge density is established by the emitter current 



p-type 

Metallurgical 
junction 



n-type 



p-type 

Metallurgical 
junction 




W B 

Emitter Base Collector 

Distance in base region — *■ 

FIG. 5-9. Stored charge in the base region for the calculation of 
diffusion capacitance. 

which flows into the base region because the emitter-to-base junction is 
forward biased. A change in the voltage across this junction results in a 
change in the minority charge in the base region, and because of this 
effect, the junction acts as if it had an incremental capacitance known as the 
diffusion capacitance. 

From Eq. (4-31), the hole density in the base region just inside the emitter- 
base depletion region (x = in Fig. 5-9) is 



P = Pno exp 



m 



SEC. 5-3 CHARACTERISTICS OF THE CURRENT FLOW ACROSS THE BASE REGION 123 

where V EB is the forward bias on this junction. Ignoring recombination, the 
charge-versus-distance curve in the base region is a straight line as shown in 
Fig. 5-9. The charge stored in the base region per unit area of the junction 
is the shaded, triangular region in Fig. 5-9 and is given by 



ePnoW 



exp(^f fi ) (5-13) 



where W is the active base width. 

If the input voltage V EB is varied, the charge stored in the base region will 
change. The value 

c ° = £ (5 - 14) 

is defined as the diffusion capacitance of the transistor per unit area of the 
junction. From Eqs. (5-13) and (5-14), 



But from Eq. (5-9), 



and so 



dQ e'PnoW leV EB \ 

Ld 'w; b - ~2kf~ exp vw) 

. eD p (eV EB \ 



eW 2 
C D = 2kTD J per Unit area ^ 5_1 ^ 



The incremental capacitance is given in Eq. (5-15) in terms of the current 
density J, and thus the total diffusion capacitance across the junction is 
proportional to the total emitter current, I E . 

The diffusion capacitance is in addition to the incremental capacitance of 
a p-n junction that exists because of the presence of the charge depletion 
regions (see Sec. 3-4). For a small transistor, the emitter-to-base diffusion 
capacitance is usually above 100 picofarads at I E = 1 milliamp whereas the 
junction capacitance is typically of the order of 10 picofarads. The diffusion 
capacitance for the collector-to-base junction is very small since there is little 
alteration in the charge in the base region as the collector voltage changes. 
The incremental capacitance across the two junctions is a major factor in 
determining the upper frequency response of a transistor. This matter is 
discussed in some detail in Chap. 10. 

5-4 The Transistor as a Device: Common Base Circuit 

Figure. 5-10 shows a method of operating a pnp transistor. This is known as 
the common base (CB) circuit. Under normal transistor operation, the emitter- 
to-base junction is operated in a forward direction with a direct current I E , 



124 



THE JUNCTION TRANSISTOR 



CHAP. 5 



and the collector-to-base junction is biased in a reverse direction with a 
current flow of I c . From the preceding analysis, I c is slightly less than I E 
and the output characteristics in Fig. 5-10 show a family of curves, each 
curve having a different value of I E . The difference between I E and I c at any 
point is the base current, I B . The output curves are nearly horizontal lines 
provided the collector-to-base junction is reverse biased. This indicates a 
high incremental output resistance of the order of 10 6 ohms. By contrast, the 
incremental input resistance of the transistor is of the order of a few ohms. 



Limiting 
resistance 



Emitter- to - 
base battery 




Zi. Collector-to- 
ry~ base battery 



/ F =0 



Collector- to -base voltage 
(reverse bias) 

FIG. 5-10. The common base (CB) circuit and the common base 
output characteristics. 



One of the principal components of this is the emitter-to-base junction 
incremental resistance (Sec. 4-7). 

In most of the circuit analyses which will be carried out in the following 
chapters, the current gain of the transistor will be the factor of greatest interest 
(i.e., the ratio of the output and input currents). The transistor is a current- 
operated device, and the voltages necessary to sustain suitable currents across 
the junctions are of secondary importance. For example, the emitter current 
versus emitter-to-base voltage curve is only rarely plotted since it is so tem- 
perature dependent (see Eq. 5-9). The small emitter-to-base voltage required 
for forward operation of the junction is readily obtained, and interest is 
chiefly centered on the value of the emitter current. 

Before a complete analysis of the operation of the transistor amplifier is 
attempted, it is of interest to show the capabilities of the device under a-c 



SEC. 5-4 



THE TRANSISTOR AS A DEVICE: COMMON BASE CIRCUIT 



125 



operation. This can be done by reference to Fig. 5-1 1 where a small alter- 
nating-voltage generator V e has been placed in series with the input circuit. 
Assuming that the alternating-voltage V e is much less than the direct emitter- 
to-base voltage, a small alternating current I e will flow in the input circuit 
given by 

Ve = IJi (5-16) 

where r t is the incremental input resistance under the given conditions. If the 
alternating current in the output circuit is I c , the alternating output voltage 
across the load is 

V = I C R L (5-17) 

The current gain of the circuit is 



A, = 



and the voltage gain is 



A„ = 



IcRl 
hr< 



(5-18) 



(5-19) 



Equations 5-16 through 5-19 apply to small-signal a-c operation. It can be 
seen, however, that the ratio of the alternating currents I c and I e is almost the 
same as the ratio of the direct currents measured under the same conditions 
since the mechanism of charge conveyance in the base region is the same at 
low frequencies as at zero frequency. When the load resistance, R L , is zero 
and the collector-to-base voltage is a constant, the current gain of the CB 
circuit is known as the a (alpha) of the transistor and has a value between 
0.95 and 0.995. This is the most important transistor parameter. 



Alternating 
current 



Alternating 
current 



v?<§ 



Alternating 
voltage 



Suitable 
emitter-to- ~^F 
base voltage 



£> 



\ 



/ 



<S> 



r l Output V 
o 



tl 



FIG. 5-11. Inclusion of a small alternating voltage in the input of 
a CB transistor circuit. 

In the case shown in Fig. 5-11, we may approximate the condition 
R L — by R L < 0.1 r , where r is the incremental output resistance, since 
then the voltage drop across R L is a small proportion of the total alternating 
voltage in the collector circuit. Using the further approximation that a = 1, 



A n = 



(5-20) 



126 THE JUNCTION TRANSISTOR CHAP. 5 

Typical values would be 

r = 1 megohm, R L = 10,000 ohms, r { = 50 ohms 
giving a, . 1^000 . 20Q 

From the foregoing simplified analysis, the CB transistor circuit is charac- 
terized by 

1. current gain just less than unity, 

2. high voltage gain, 

3. low input resistance, 

4. high output resistance. 

5-5 The Transistor as a Device: Common Emitter Circuit 

The common emitter (CE) circuit is the most used type of transistor amplifying 
stage since it provides a high current gain. To understand its principle of 

operation consider the three direct currents 
h+. 7 c x flowing in the leads of a transistor as defined 
\. / by Fig. 5-12. 



Neglecting any charge storage effects in 
the device which might occur under transient 
conditions, 

h = h + h (5-21) 

FIG. 5-12. Assigned directions The total collector current can be represented 

of direct currents flowing in a by 

transistor. r r , r /c tvv 

I c = aI B + Icbo (5-22) 

Here the collector current is written as the sum of the small cut-off current 
which flows even when I E = (see Fig. 5-10), and alpha times the emitter 
current I E . The quantity a was defined for alternating-current conditions but 
will also apply to the direct-current case without appreciable error when the 
collector-to-base voltage is constant. Eliminating I E between Eqs. (5-21) and 
(5-22) gives 

h = r^— h + -r^ (5-23) 

I — a. 1 — a 

Now assume that a small alternating current I b is impressed upon the 
direct current I B . Assuming I CBO is a constant, Eq. (5-23) can be written as 

(ic + to) = r 1 - Vb + h) + r^ (5-24) 

1 — a 1 — a 



SEC. 5-5 



THE TRANSISTOR AS A DEVICE: COMMON EMITTER CIRCUIT 



127 



Subtracting Eq. (5-23) from Eq. (5-24) 



/, = 



1 - 



(5-25) 



The current ratio IJI b is the alternating-current gain of a transistor circuit 
when the base lead is used as the input and the collector lead is the output. 
Such an arrangement is known as a common emitter (CE) circuit and is 
shown in Fig. 5-13. This figure also shows the output characteristic curves 



Alternating 
voltage 
generator 
Suitable 
base-to- 
emitter voltage 



<§ 



Alternating 
current 



<i> 



Alternating 
current 



<& 



Output 
— o 



£T 




Collector-to-emitter voltage 
(reverse bias) 

FIG. 5-13. The common emitter (CE) circuit and the common 
emitter output characteristics. 



for this connection. By comparing the characteristic curves for the CE and 
the CB connection it is seen that the output incremental resistance of the CE 
circuit is much lower. This is a factor in the design of CE stages such as the 
one given in Fig. 5-13, since, to apply Eq. (5-25) directly, R L must be much 
less than the incremental output resistance. Suitable values for the CE stage 
are 

r = 20,000 ohms, R L = 1500 ohms, 

r { = 1000 ohms, a = 0.98 

Assuming R L « r , the current gain of the stage is 

h 1 - a 



128 THE JUNCTION TRANSISTOR CHAP. 5 

The p (beta) of a transistor is the maximum current gain that can be obtained 
from the CE circuit and is commonly between 20 and 200. The acceptable 
limits of j3 for a particular type of transistor will be specified by the manu- 
facturer. For the transistor mentioned above, p = 49 and so the current gain 
of the circuit assuming R L « r is 

A t = p = 49 

The voltage gain through the stage is given approximately by 

hR k _PR k _ 49(1500) _ 
1 l b r { r t 1000 ~ ,XD 

Thus the CE transistor stage is characterized by 

1 . current gain fairly high (between 20 and 200), 

2. voltage gain fairly high, 

3. fairly low input resistance, 

4. fairly high output resistance. 

The CE circuit is the most used amplifier because it has a useful current gain 
and individual stages can be cascaded to form a multistage amplifier. 

5-6 The Manufacture of Transistors 

A. Materials. Germanium was the first material to be used in the manu- 
facture of transistors, but in recent years increasing use has been made of 
silicon. Silicon has a higher forbidden gap width (1.1 volts for silicon, 
0.72 volts for germanium) and so silicon devices can be operated at higher 
temperatures (about 175° C for silicon and 75° C for germanium). The wider 
band gap of silicon leads to lower leakage currents and higher impedances. 
Silicon has a sharper avalanche property and hence allows fabrication of 
voltage regulator diodes (see Sec. 3-6). In addition, silicon junctions can be 
made to operate at higher voltages. Finally, silicon has a remarkable natural 
oxide which allows easy fabrication of diffused and surface passivated devices 
(see Sec. 9-10). 

A single crystal of germanium or silicon must be formed as a step in 
transistor manufacture. This crystal is grown by slowly withdrawing a 
"seed" crystal having the desired orientation from a crucible containing the 
molten semiconductor. Although chemically pure material is used, the small 
amount of group 3 or 5 impurity still present in the semiconductor determines 
its conductivity. Removal of these impurities is obtained by zone refining. In 
this technique, molten zones, obtained by localized heating, are moved along 
a semiconducting rod, and it is found that the impurities (being more soluble 
in the melt than in the solid) pass along to one end, leaving the rest of the rod 
with a reduced impurity content. Purity of better than 1 part in 10 10 may be 



SEC. 5-6 THE MANUFACTURE OF TRANSISTORS 129 

obtained with this method. It is more difficult to produce pure single crystals 
from silicon than from germanium but most of the technical problems 
associated with the use of silicon have now been overcome. 

B. Point-contact transistors. Many semiconducting diodes and some early 
transistors were made by pressing fine cats-whisker contact wires on to semi- 
conducting wafers. Many diodes are still made in this way, but transistor 
production of this type has now ceased. The three major drawbacks to the use 
of point-contact transistors were their tendency to instability, their high noise 
level and the difficulty of predicting their characteristics. 

C. Grown-junction transistors. This type of device accounts for a few per cent, 
of the total output of the industry. A typical germanium npn grown-junction 
transistor is cut from a bar of up to 1 in. in diameter. The single crystal bar is 
grown from a germanium melt containing arsenic to make it n-typz. During 



Collector 



p-type 

alloy region^ ^^^^^>^' IndlUm 




p-type alloy region- 7 J\ ^Active base region 

Indium 



FIG. 5-14. Schematic diagram of a typical alloy-junction tran- 
sistor. 

the crystal growing process, gallium is added to the melt to make it /?-type. 
Then, more n-typo, dope is added to make an npn sandwich. The bar is then 
sliced longitudinally into small sections about 0.025 x 0.025 x 0.125 in. 
long. The/7-type base region is about 1/1000 in. thick. This device is suitable 
for audio-frequency operation only. 

D. Alloy-junction transistors. Many transistors are now made using a structure 
similar to Fig. 5-14. Here an n-type semiconducting wafer of 0.08 x 0.08 x 
0.004 in. thick serves as the «-type base. Indium is placed on the crystal 
wafer, and is heated to a carefully controlled temperature so that the indium 
dissolves (by alloying) the germanium as shown in Fig. 5-14. Upon cooling, 
two p-type regions regrow from the indium-germanium alloy and form the 
emitter and the collector regions. Both junctions are of abrupt or step-like 
nature. By alloying further into the n-type material, the thickness of the base 
region can be reduced. In this way, the frequency response of the device can 
be increased at the expense of reducing the allowed operating voltage to 
prevent breakdown. The manufacturing techniques become difficult when the 



130 THE JUNCTION TRANSISTOR CHAP. 5 

base thickness is reduced below a certain value and so the frequency response 
of the CB circuit is limited in practice to tens of megacycles per second 
which is adequate for many purposes. A variation of this technique involves 
chemically etching the base region to reduce the base thickness before 
alloying takes place. The device is then known as a surface-barrier transistor. 

E. Diffusion techniques. This method of transistor and diode manufacture is 
increasingly being used in the semiconductor industry. The diffusion referred 
to here (doping impurity diffusion) occurs at high temperatures when there is 
atomic migration of group 3 or 5 material into the semiconductor. This pro- 
cess gives accurately controlled penetration depths, and base thicknesses of 
less than 5 x 10 ~ 5 in. can be produced as the difference in depth of two 



n-type diffused emitter 
Base contact 



region 6xl0 _5 in. 




Emitter contact 
Active base X»l JL .p-type diffused region 



^U- Original ra-type 
"|| material 
I (collection) 



X. 



Ohmic contact 



FIG. 5-15. Cross section of a mesa diffused-base transistor 
(simplified). 

diffusion cycles. Junctions produced by this method are not abrupt but graded 
in some way, giving lower capacitances and often higher allowable operating 
voltages. Diffusants are usually in vapor form, and phosphorus (/2-type), 
boron (p-type) and gallium (/?-type) are commonly used. In general, hun- 
dreds of diffused devices can be made on wafers of \ to 1 in. diameter and 
then be scribed into several hundred individual devices. In this way, a highly 
uniform product of low cost may be obtained. A common manufacturing 
technique is described below. 

The diffused-base "mesa" transistor is shown in simplified form in 
Fig. 5-15. The word "mesa" means "a flat-topped rocky hill with steeply 
sloping sides, common in the southwestern U.S." and describes the form of 
this device. The square wafer of /7-type material shown is one of 100 to 
1000 units which are made simultaneously on the same semiconducting slice. 
After cleaning and polishing of the top face of the slice, boron is diffused 
into the material to a depth of 1.6 x 10" 4 in. to form a /?-type layer. Then 
a second diffusion of phosphorus is made over a controlled region to a 
depth of 1 x 10 ~ 4 in. to give abase region of thickness 6 x 10 " 5 in. between 
the first and the second diffusions. Emitter and base metallic contacts are 
now evaporated on to the surface using a precision mask, and unwanted 
material is etched away to give the transistor its "mesa" shape. The semi- 



SEC. 5-6 THE MANUFACTURE OF TRANSISTORS 131 

conducting slice must still be cleaned and broken into its individual pieces 
before mounting and lead attachment takes place. The expense incurred in 
this construction is offset by the large quantity and uniformity of transistors 
produced from one slice of semiconductor. Diffused transistors are capable 
of operation above one kilomegacycle per second. 

Transistors may be made using both diffusion and alloying technologies. 
One manufacturing process starts with a high resistance n-typc wafer into 
which is diffused an n-type skin. An indium dot is next alloyed to one face to 
produce a p-n emitter to base junction. On the reverse face of the wafer, the 
«-type skin is removed to uncover the near-intrinsic «-type material below, 
and to this is alloyed an indium collector. This method of fabrication pro- 
duces a pnp transistor with a graded base having the high resistance part of 
the base close to the collector. Thus a small collector-to-base capacitance 
results, and there is also a "built-in" electric field which sweeps the injected 
minority carriers across the base region to the collector (see Sec. 9-8). Fre- 
quency responses of greater than one kilomegacycles per second are obtainable 
but power dissipation is generally limited to below one watt and the cost of the 
device is high. 

BIBLIOGRAPHY 

Junction transistors 

DeWitt, David, and Arthur L. Rossoff, Transistor Electronics, New York: 
McGraw-Hill Book Company, Inc., 1957 

Gartner, Wolfgang W., Transistors: Principles, Design, and Applications, 
Princeton: D. Van Nostrand Company, Inc., 1960 

Greiner, R. A., Semiconductor Devices and Applications, New York: McGraw- 
Hill Book Company, Inc., 1961 

Lo, Arthur W., Richard O. Endres, et al, Transistor Electronics, Englewood 
Cliffs, N.J.: Prentice-Hall, Inc., 1955 

Middlebrook, R. D., An Introduction to Junction Transistor Theory, New York : 
John Wiley & Sons, Inc., 1957 

Manufacture of transistors 

Anderson, A. Eugene, "Transistor Technology Evolution," Western Electric 

Engineer, 3 (July 1959), 2-12, 3 (October 1959), 30-36, 4 (January 1960), 

14-19 

Shive, J. N., Properties, Physics and Design of Semiconductor Devices, New 
York: D. Van Nostrand Company, Inc., 1959 

PROBLEMS 

5-1 The p-n junction that was used as an example in Sec. 4-4 is now the 
emitter-to-base junction of a pnp transistor. If the base thickness is 



132 THE JUNCTION TRANSISTOR CHAP. 5 

0.002 cm, the emitter-to-base voltage is 0.2 volts in the forward direc- 
tion and the collector-to-base junction is reverse biased, find the 
current density of holes across the base region. 

5-2 Calculate the emitter efficiency for the transistor in the previous 
question. How can the emitter efficiency be improved? 

5-3 In an alloy junction transistor, the base is a field-free region under small- 
current conditions. Discuss how the current passes from emitter to 
collector showing what conditions must be fulfilled for transistor action. 

5-4 The transistor of Prob. 5-1 is operated with a collector-to-base reverse 
voltage of (a) 0.5 and (b) 20 volts. What are the collector currents in 
the two cases if V EB is a constant? Assume that the base and collector 
conductivities are identical. 

5-5 Compute the diffusion capacitance for the transistor of Prob. 5-1 
when I E = 1 ma. 

5-6 Identify the nature of the carrier currents in each region of an alloy 
junction pnp transistor. In what manner are these currents sensitive 
to (a) temperature variations, (b) bias variations ? 

5-7 A transistor with an alpha of 0.97 is operated in a common base circuit 
with a resistance of 2500 ohms in the collector circuit. A small-signal 
a-c source is connected in series with the emitter lead. The a-c output 
resistance of the transistor is 1 megohm and the input resistance is 100 
ohms. What are the a-c voltage and current gains for this circuit? 
Suggest a use for the common base circuit. 

5-8 A transistor with an alpha of 0.99 is operated in a common emitter 
circuit. What is the maximum alternating-current gain that can be 
achieved? In this circuit, the direct current, I B , is held constant as the 
temperature of the device changes. I CBO , however, is found to double its 
value for a 10° C increase in temperature. If I B = 10 /xa (constant) and 
I CBO = l ^ at 25° C, what is the value of I c at (a) 25° C, (b) 35° C, (c) 
50° C ? (Note the serious effect of changes in I CBO when the CE circuit 
is operated with constant 7 S ; this effect is analyzed in Chap. 8.) 

5-9 What are the effects of the conductivities of the three transistor regions 
on current gain, emitter efficiency and base width modulation? 

5-10 It is required to cascade two transistor stages to form a multistage 
transistor a-c amplifier. Taking the typical stages given in Sees. 5-4 and 
5-5 as a guide, design a suitable circuit for coupling (a) two common 
base stages, (b) two common emitter stages. Compute the current gain 
in both cases and discuss the results. 

5- 1 1 Describe the construction of an alloy junction and a diffused-base mesa 
transistor. By reference to current transistor literature, determine the 
capabilities of both types of transistor. 



PART 



2 



Transistors 

at 

Low Frequencies 

and under 

d-c Conditions 



6 



The 

Transistor 

as a 

Two Port 

Network 



In this chapter will be found a discussion of three 
small-signal parameter representations of the tran- 
sistor. These are the z or impedance parameters, the 
y or admittance parameters and the h or hybrid 
parameters. Of these three, the h parameters are 
most commonly used, and they will be employed for 
circuit analysis at low frequencies in Chaps. 7 and 
8. The equivalent circuits for these representations 
are given together with the hybrid-n representation 
which is used in Chaps. 9 and 10. The chapter 
opens with a general discussion of the properties of 
electronic amplifying devices. 



36 



THE TRANSISTOR AS A TWO PORT NETWORK 



CHAP. 6 



6-1 General Considerations of Electronic Devices 
as Amplifiers 

There are four requirements which must be fulfilled if an electronic device is 
to be a useful general-purpose amplifier. In the first place, it must have a 
considerable voltage or current gain and be able to control the power in its 
output circuit with the expenditure of relatively little power in the input cir- 
cuit. Secondly, the device must be linear to prevent the introduction of un- 
wanted distortion. When used as a power amplifier, a third requirement is 
that this linearity must be obtainable over a substantial part of the output 
characteristics of the device. The last condition is that it must be possible to 



Y 



generator 



/ 



(a) 



£T 



generator 



fr 



(b) 



generator 



(c) 

FIG. 6-1. Three transistor connections: (a) CB circuit; (b) CE 
circuit; (c) CC circuit. 

cascade the device to form a multistage amplifier. Triode and pentode vacuum 
tubes meet all these conditions and any competing device must measure up to 
these requirements before other points of comparison are made. It will be 
shown that the transistor is capable of providing a high gain with a substantial 
amount of linearity in a cascaded amplifier, and so it can be used as a 
general-purpose amplifying device. 

Figure 6-1 shows a transistor arranged in three possible configurations. 
Two of these, the CB and the CE circuits, have been briefly discussed in 
Sees. 5-4 and 5-5, and the third diagram shows the common collector (CC) 
circuit. Of these three circuits, only the CE stage can be used in multistage 
amplifiers. The reasons for this will be given in Chap. 7. 

The transistor is mainly used as an amplifier of alternating currents, but 



SEC. 6-1 



GENERAL CONSIDERATIONS OF ELECTRONIC DEVICES AS AMPLIFIERS 



137 



suitable direct-current operating conditions must also be established. This 
may be done by plotting the static characteristics of the device. Figure 6-2 
shows that the transistor in any one of its three connections can be regarded 
as a two port device linking the input and output circuits. The symbols used 
in this figure define the instantaneous total current and voltage at the input and 
output terminals. These symbols, which conform to current practice, there- 
fore apply when direct-current and voltage measurements are taken on the 
transistor to give the input and output static characteristic curves. The 
subscripts e, b, and c will be used where appropriate to denote values at the 
emitter, base, and collector terminals. 

Consider Fig. 6-3 which shows the static input and output characteristics 
for the CE and CB circuits respectively. These curves are obtained from a 
measuring circuit of the type shown in Fig. 6-4 which applies specifically to 
the CE connection. As we can predict from semiconductor theory, useful 
operation occurs when the base-to-emitter junction has a forward voltage 
bias and when the collector-to-base junction has a reverse-applied voltage. 
We note that the reverse bias between the collector and the base is responsible 
for a large output resistance, whereas the forward-biased, emitter-to-base 
junction has a low resistance. Since the emitter current and the collector 
current are essentially the same, current is transferred from a low to a high 
resistance circuit, leading to the name trans-resistor or transistor. 



Input 
circuit 



l l 










*2 














Transistor 






Vl 




O 




^2 


o 










o 



Output 
circuit 



FIG. 6-2. The transistor as a two port device linking output and 
input circuits. Instantaneous total voltages and currents are 
shown. 



Valuable information can be obtained from the input and output static 
characteristic curves of any electron device. Voltage or current gain can be 
predicted by inspection of the family of output curves. Linearity of gain over 
a wide range of input and output currents can be assessed by checking the 
constancy of the spacing of the output curves for equal variations in the input 
parameters. Wide range of operation is suggested by an extensive region over 
which the output curves have a near-constant slope. The characteristics for 
the CE connection of the transistor show that this method of operating the 
device will give a high current amplification since a small variation in i b pro- 
duces a large variation in i c . The spacing between the output curves is 
approximately constant for equal variations in input current showing that the 
CE connection will give substantial linearity. The operating range is plainly 



138 



THE TRANSISTOR AS A TWO PORT NETWORK CHAP. 6 




v cb = - 1 volt 
(reverse bias) 



0.1 0.2 

v eb in volts (forward bias) 

Input curve 





10 


_ i c = 10 ma 


C8 

£ 
a 


o i e = 8 ma 


6 


i e = 6ma 


"» 






4 


L = 4 ma 




2 


i e = 2 ma 






i e = ma 



(a) 



v ch in negative volts (reverse bias) 
Output curve 




v ce = - 10 volts 
(reverse bias) 



0.1 0.2 0.3 

v be in volts (forward bias) 
Input curve 



(b) 




2 4 6 8 10 

v ce in negative volts (reverse bias) 
Output curve 



FIG. 6-3. Input and output characteristics of pnp germanium 
transistor, (a) CB connection, (b) CE connection. 




FIG. 6-4. A circuit suitable for taking static characteristic curves 
for the CE connection. 



SEC. 6-1 GENERAL CONSIDERATIONS OF ELECTRONIC DEVICES AS AMPLIFIERS 139 

seen from the shape of the curves. It may be concluded, therefore, that the 
CE stage is a useful current amplifier. However, inspection of the base-to- 
emitter characteristic shows marked nonlinearity which must be taken into 
account in amplifier design. 

The CB and the CC characteristics may also be analyzed in a similar 
manner. For the CB stage, it will be seen that a current gain of slightly less 
than unity will result, and so this stage is unsuitable for cascading in a multi- 
stage current amplifier. The CC static characteristics (which are not shown 
here) indicate that a high current gain can be obtained when one stage only is 
used, but, in Chap. 7, it is shown that this circuit, too, is never employed in 
multistage amplifiers. 

The characteristics shown in Fig. 6-3 are the most useful ones in practice. 
We have reproduced the output curves with input current as a running para- 
meter since it is known by experience that the transistor is best regarded as a 
current amplifier. The output curves drawn with input voltage as a parameter 
plainly show the nonlinearity which exists between input voltage and output 
current. 

The design of single and multistage amplifiers prior to the advent of tran- 
sistors had been dominated by considerations which applied to vacuum tubes. 
Loudspeakers, microphones, etc., had been made with impedances and 
power requirements suitable for triode and pentode circuits. Furthermore, 
matching between stages was considered solely from a voltage standpoint 
since the vacuum tube is primarily a voltage amplifier. However, the tran- 
sistor is a current-operated device. It is the current gain per stage which 
determines the properties of a multistage transistor amplifier and the voltage 
gain has a much smaller significance. Consequently, since about 1950, the 
design of electronic equipment has been studied with regard to the use of both 
voltage and current amplifying devices and this has resulted in a better un- 
derstanding of the processes involved. 

6-2 Alternating-current Operation of the Transistor 

In this, and the following chapter, only the alternating-current operation of 
the transistor will be considered. It will be assumed that the transistor is 
operating with a constant emitter, base, and collector current such that 
"small-current" conditions hold (see Sec. 4-2). The selection of a suitable 
operating point on the characteristic curves is discussed in Chap. 8. 

There are four variables in the two port representation of a device given 
in Fig. 6-2. These are the two voltages i^and v 2 and the two currents i x and 
i a . The graphical relationship between these quantities is shown by a pair of 
the static characteristic curves of Fig. 6-3 which apply to the two port re- 
presentation of the particular connection chosen. These curves determine the 
potentialities of the device, as described in the previous section, but for a-c 
analysis, it is better to consider the mathematical relationship between the 



140 THE TRANSISTOR AS A TWO PORT NETWORK CHAP. 6 

four variables. In general, an amplifier is operated in the region where the 
curves are straight lines so the equations should be linear. 

There is some function, f a , relating one of the variables, say v u with the 
other three. Thus 

Vi = f a (v 2 , h, i 2 ) (6-1) 

The quantity, v 2 , could have been written in a similar general form with a 
different function, f b , involving v l9 i lt and / 2 , hence 

^2 = f b (v lf h, i 2 ) (6-2) 

From Eqs. (6-1) and (6-2), we can write 

Vi = /iO'i, i 2 ) (6-3) 

v 2 = f 2 (h, * 2 ) (6-4) 

where /x and/ 2 are two functions which can be found if the functions f a and 
f b are known. Equations (6-3) and (6-4) show that i x and i 2 can be considered 
as independent variables. 

By a similar analysis, the two currents can be written in terms of the two 
voltages to give 

k = fa(v u v 2 ) (6-5) 

k = U(vi, v 2 ) (6-6) 

There is no reason why one current and one voltage cannot be taken as the 
independent variables. The curves of Fig. 6-3 suggest the equations 

Vi = f 5 0'i, v 2 ) (6-7) 

h = feO'i, ^2) (6-8) 

Further equations can be written using the three remaining pairs of in- 
dependent variables, but they serve no useful purpose in our analysis. 

The relationships given above apply to the instantaneous total voltages 
and currents. For small variations in these quantities about the operating 
point, Eqs. (6-3) and (6-4) become 

dVl = ^ dil + ^l d i 2 (6-9) 

Oil ^h 

dv 2 = d pdi 1 + d pdi 2 (6-10) 

Cl x Cl 2 

Small-signal operation applies when the variations are restricted to such a 
small amplitude that the partial differential terms in Eqs. (6-9) and (6-10) can 
be considered as constant. Inspection of the static characteristics shows how 
wide a range this will be. In general, this linear region should extend to at 
least 10 per cent of the corresponding direct operating values or the device 



SEC. 6-2 ALTERNATING-CURRENT OPERATION OF THE TRANSISTOR 141 

would not be used. Small-signal operation is usually taken to mean the con- 
ditions where all four partial differentials are constant. Linear amplification 
can often be obtained over a much wider range if one or more of the partial 
differential coefficients can be ignored. 

The four partial differential terms have the dimensions of impedance and 
can be written 

Zll = ^' Z ^ = W 2 (6 ~ n) 

dv 2 _ dv 2 

z 21 = -^ z 22 - — 

In practice, the transistor will frequently be used for the amplification of 
sinusoidal currents. Hence, it is convenient to replace the changes in in- 
stantaneous total voltage and current by rms a-c quantities which are con- 
sidered to be impressed on the direct values. Equations (6-9) and (6-10) may 
now be written as 

V x = z lx I x + z 12 / 2 (6-12) 

V 2 = z 2 Ji + z 22 I 2 (6-13) 

The four impedance terms can be defined as follows: 



Z,o = 



ZoA = 



Vi 
h 

Vi 

h 

Yi 
h 

u 



/ 2 =o 



/2=0 



and is the input impedance when the output is open- 
circuited to a-c. 

and is the reverse transfer impedance with the input open- 
circuited to a-c. 

and is the forward transfer impedance with the output 
open-circuited to a-c. 

and is the output impedance with the input open-circuited 
to a-c. 



h 

The open-circuit a-c condition assumes that the direct currents and voltages 
in the two port network are undisturbed. Practical methods of achieving this 
are discussed in Sec. 6-7. 

Equations (6-12) and (6-13) are the defining equations of the open-circuit 
impedance representation of the linear two port (four terminal) network. The 
four impedance parameters, z n , z 12 , z 21 , and z 22 will be used to synthesize a 
circuit which represents the device within the " black box " of Fig. 6-1 as far as 
small-signal a-c conditions are concerned. This is the z parameter represen- 
tation. 

If the second pair of independent variables is considered, Eqs. (6-5) and 
(6-6) can be written in the form 

h =J>iiKi +^12^2 (6-14) 

h = y2iVi + y 22 v 2 (6-15) 



142 



THE TRANSISTOR AS A TWO PORT NETWORK 



CHAP. 6 



where the y parameters are defined below 



^11 = 



yxi 



y 2 i 



y22 = 



v 2 = o 



Vl=0 



v 2 = o 



Vi=0 



and is the input admittance when the output is short- 
circuited to a-c. 

and is the reverse transfer admittance with the input short- 
circuited to a-c. 

and is the forward transfer admittance with the output 
short-circuited to a-c. 

and is the output admittance with the input short- 
circuited to a-c. 



Here the short-circuit a-c conditions must be achieved without altering the 
direct currents and voltages (see Sec. 6-7). These are the short-circuit ad- 
mittance parameters of the linear two port network and can be used to syn- 
thesize an equivalent circuit for the device under small-signal a-c conditions. 
This is the y parameter representation. 

The third, and last, representation which will be given here uses the hybrid, 
or h, parameters. These parameters arise from considering Eqs. (6-7) and 
(6-8) and are defined by the two equations below. 



V x = h 11 I 1 + h 12 V 2 



h = /*2lA + ^22^2 



(6-16) 
(6-17) 



where 



and is the input impedance when the output is short- 
circuited to a-c. 

and is the reverse voltage amplification factor with the 
input open-circuited to a-c. 

and is the forward current gain with the output short- 
circuited to a-c. 

and is the output admittance with the input open-circuited 
to a-c. 

The parameter /* n has the dimensions of impedance, and it is equal to 1/yu. 
h 22 has the dimensions of admittance and is equal to \/z 22 . h 12 and h 21 are 
nondimensional. It will be shown that h 21 is of great importance in a transistor 
since it is the current gain of the device in the short-circuit condition. 



h 11 


Vi 

' h 


h 12 


Vi 

v 2 


h 2 i 


h 


h 2 2 


h 

v 2 



v 2 =o 



h = o 



v 2 = o 



6-3 z, y, and h Parameters: General Comments 

So far we have defined three sets of parameters to represent the two port 
network. These representations apply to small-signal operation where a 
linear relationship exists between the alternating currents and voltages. Since 



SEC. 6-3 



z, y, AND h PARAMETERS: GENERAL COMMENTS 



143 



all three representations refer to the same device, the three sets of parameters 
must be interrelated. At first sight, it is difficult to see why three representa- 
tions of the same device are required. The reasons are partly historical and 
partly technical; moreover, the various manufacturers continue to specify 
their transistors in different ways. Historically, the z parameters were used a 
great deal in the early days of transistor electronics. This representation was 
simple and direct since an impedance was associated with each of the tran- 
sistor leads. However, measurement of the z parameters is difficult (Sec. 6-7), 
and the representation is less useful at high frequencies. The y and h para- 
meter representation gained in popularity with increasing usage of the tran- 
sistor. In this book, the h parameter representation will be used for the 




FIG. 6-5. Input and output circuits connected to the two port 
"black box." Rms a-c voltages and currents are shown. 



analysis of low frequency operation of the device. In Part 3, where high 
frequency and pulse operation are considered, a modification of the y para- 
meter representation is employed which is known as the hybrid-n circuit, and 
it will be referred to later in this chapter. 

In the next three sections, equivalent circuits will be developed for the z, 
y, and h parameter representations respectively. In general, the justification 
for using these circuits will be left to the reader since this is standard two port 
network theory. The references given at the end of this chapter deal with 
this topic in detail. 

The object of using any of the three sets of parameters is to provide a 
simple circuit representation of the transistor. Thus a transistor stage may be 
analyzed by replacing the device by an equivalent two port network contain- 
ing resistances, capacitances, inductances, and generators. Figure 6-5 shows 
input and output circuits connected to the "black box." We require to know 
the following quantities under all circuit conditions. 



Input impedance, 


z« 


h 


Output impedance, 


Z„ 


v 2 

h 


Current gain, 


A, 


-k 

T 



144 THE TRANSISTOR AS A TWO PORT NETWORK CHAP. 6 

y 
Voltage gain, A v = ~ 

Power gain, A p = A { A V 

These terms are known as performance quantities, and the aim of the following 
circuit representation and analysis is to determine these quantities under any 
conditions which may exist. The voltage generator, V L , is rarely connected in 
practice, but it is included in Fig. 6-5 to simplify the calculation for the output 
impedance. 

6-4 Equivalent Circuit using z Parameters: the T Circuit 

Consider the defining equations of the z parameter representation. 

Vi = Z1J1 + z 12 / 2 (6-12) 

^2 = Z 21 h + Z 22 / 2 (6-13) 

It is found that there is a simple network of impedances given in Fig. 6-6 
which represents a linear two port network when z 12 = z 21 . This is known as 
a passive network. Mesh analysis of Fig. 6-6 justifies the use of the circuit. 



Z\\- z \2 2 22 _2 12 

-I I 



v, 



*12 



FIG. 6-6. The T circuit for a passive network. 

A passive circuit can be defined as one which has no energy generating source. 
This circuit is not adequate for the transistor, however, since measurements 
show that z 12 ^ z 21 . The transistor must, therefore, be represented by an 
active circuit which contains at least one generator in addition to the im- 
pedances. Rewriting the circuit equations by adding and subtracting z 12 h to 
the right-hand side of Eq. (6-13), we have 

V x = z lx I x + z 12 / 2 (6-12) 

V 2 = z 12 h + z 22 I 2 + (z 21 - z 12 )I lt (6-18) 

Equations (6-12) and (6-18) now relate to the network of Fig. 6-7 as can be 
shown by simple circuit analysis. The term (z 21 — z 12 )/i accounts for the 
constant voltage generator in the circuit. When z 12 = z 21 , this generator 
vanishes to give the circuit of Fig. 6-6. 



SEC. 6-4 



EQUIVALENT CIRCUIT USING z PARAMETERS: THE T CIRCUIT 



145 



z \\ ~ 2 12 



2 22 ~ 2 12 
- I I - 



(z 2 i-z l2 )h 

— 0_^> 



*12 



FIG. 6-7. The active T circuit. 

The circuit of Fig. 6-7 may be applied directly to any of the three con- 
nections of the transistor given in Fig. 6-1. When the CB configuration is 
selected, the three impedances may be identified with the three leads of the 
transistor as shown below. At low frequencies only the resistive components 
of the impedances are important and so : 



Z 22 
*21 



Z 12 ~ r e-> 

z 12 — r bi 

Z 12 = r a 

Z 12 = r m 



the emitter resistance 

the base resistance 

the collector resistance 

the transresistance 

Typical values for an alloy junction transistor are found to be : 

r e = 20 ohms 

r b = 500 ohms 

r c = 1 megohm 

r m = 0.98 megohms 

and the circuit is given in Fig. 6-8(a). From this figure, the numerical value 
of the current gain is 



v 2 = o 



o- ^yvv-j-^vvv-Q — -p 

v, < v 2 



' m r t 



AA/V 



Vx 



r. 



(a) 



(b) 



FIG. 6-8. Two common T equivalent circuits for the CB con- 
figuration. 



146 THE TRANSISTOR AS A TWO PORT NETWORK CHAP. 6 

since r e « r b « r c . The short-circuit current gain of the device in this con- 
nection is known as a (alpha) for the transistor. For the values given above, 
a = 0.98. Figure 6-8(b) shows an exactly equivalent circuit where the series 
(Thevenin) resistance-voltage generator combination of r c and r m /i has been 
replaced by a parallel (Norton) resistance-current generator combination 
formed by r c and hr m lr c . Both of these circuits will be found in transistor 
literature. The input current I y in the CB representation is the alternating 
emitter current, and the output current I 2 is the alternating collector current. 
These currents are defined as positive when flowing into their respective 
terminals. This is opposite to the direction of current flow which is commonly 
assumed in Norton and Thevenin equivalent circuits, and so the sign of the 
current term must always be taken into account. 

A r T ?2 



Vi 



w\ 1 v\a (Jhr 



r.f 



■O 



FIG. 6-9. Rearrangement of CB circuit of Fig. 6-8 (a) to make 
a CE circuit. 

The circuits of Fig. 6-8 are seen to be a logical development of the theory 
given in Part 1 . It was shown there, that the emitter current passes across the 
emitter-to-base junction, through the base region and into the collector. 
Either of the arrangements fits into this pattern. However, a difficulty arises 
since the CB circuit is rarely used as an amplifier because its current gain is 
less than unity. In practice, the CB circuit is retained to identify the values 
r e , r b , r Ci and r m , but the circuit of Fig. 6-8(a) is rearranged to give the CE 
circuit of Fig. 6-9. Note that the conventional symbols V u V 2 , I lt and / 2 
have been retained for the two port network, but now the input is between 
base and emitter and the output is taken from collector to emitter. To con- 
form with Fig. 6-8(a), the current term controlling the constant voltage 
generator is now I e not I x . 

The performance quantities defined in Sec. 6-3 can be calculated for the 
CB configuration by substituting either circuit of Fig. 6-8 in place of the 
"black box" of Fig. 6-5. For the CE circuit, Fig. 6-9 can be used. A further 
rearrangement of the active T circuit is required for the CC configuration. 
The resulting circuit analysis for any of the three cases is tedious but not 
difficult. The results of this analysis are given in Table 6-1. These values 
have been calculated at low frequencies where the impedances can be con- 
sidered as resistances. 



Q0_ 
I 






4) 0) *>• 



g £ 

£ 3 tf 



13 OS 

£ .S 



12 C 



Ph -5 



i '3 

j u 
< c 



3 5 

II 



c 



a o 

3^ 



60 


60O 


a> 


c II 


60 


ctf 


<L> 


w 3 


5* 


> 


U 



48 



THE TRANSISTOR AS A TWO PORT NETWORK 



CHAP. 6 



6-5 Equivalent Circuit using y Parameters: the n Circuit 

Equations (6-14) and (6-15) which define the y parameters are repeated here 
for convenience. 

h =ynV 1 + y 12 V 2 (6-14) 

h = y2iVi + y 22 v 2 (6-15) 

Figure 6-10 shows the passive -n circuit which applies when y 12 = y 21 . 



h 



V, 



-y\z 



yz2+y\2 



FIG. 6-10. The it circuit of a passive network. 

The passive circuit needs only admittance elements in its network representa- 
tion. Adding and subtracting y 12 V 1 to Eq. (6-15) and rewriting Eq. (6-14), 
we have a pair of equations relating to the active circuit. 



h = ynVi + yi 2 v 2 

h = yi 2 Vi + ^22^2 + (y 2 i 



yi 2 )v 1 



(6-14) 
(6-19) 



An equivalent network of the active circuit is given in Fig. 6-11. The 
term (y 21 — ^12)^1 is represented as a current generator in parallel with the 
output terminals, and this circuit is sometimes used to represent a transistor. 



-yw 



yw +yu 



^22+^1 2 Qfoi-yuW. 



FIG. 6-1 1. An active -n circuit. 

The hybrid-TT circuit is a useful representation of the CE connection of the 
transistor at high frequencies. This circuit is shown in Fig. 6-12. It is an 
active tt circuit with an added resistance in the input lead. This resistance, 
which is shown between terminals B and B\ is known as the base spreading 
resistance and is the ohmic resistance of the semiconducting material between 
the base connection and the active part of the base, B'. The hybrid-^ circuit 
is considered in detail in Chaps. 9 and 10, where it is shown to be a good 
representation of the transistor at high frequencies. 



SEC. 6-6 EQUIVALENT CIRCUIT USING h PARAMETERS 



149 



Base spreading 

resistance R 

Bo W\ 



EO- 



AA/V 



oC 



8 



gmVfe 



-o£ 



FIG. 6-12. The hybrid-7r circuit. 

6-6 Equivalent Circuit Using h Parameters: 
a Two Generator Representation 

The third set of parameters are the ones which are most used for low fre- 
quency analysis at the present time. These are the hybrid or h parameters as 
defined by Eqs. (6-16) and (6-17). 

Vi - Aii/i + h 12 V 2 (6-16) 

h = h 21 I 1 + h 22 V 2 (6-17) 

where h xl is an impedance, h 22 is an admittance, and h l2 and h 21 are non- 
dimensional, 



h u 



Q^ 1 



i h 2 \ h 



FIG. 6-13. h parameter equivalent circuit. 

There is a two generator, two branch representation which fits the two 
circuit equations. This is given in Fig. 6-13. Equation (6-16) is seen to be 
the mesh equation of the input circuit and Eq. (6-17) is the nodal equation 
of the output circuit. The input contains an impedance, h lx , and a voltage 
generator, h 12 V 2 , in series and is thus the Thevenin equivalent of the device as 
seen from the input terminals. The admittance, h 22 , and the current generator 
h 21 I x comprise the Norton equivalent looking into the output terminals. 

There are several reasons why h parameters are preferred in transistor 
circuitry over the other representations. As shown in the next section, they 
are easier to measure. They also readily fit in with the usual static charac- 
teristic curves as explained previously. h 21 , for instance, represents the short- 
circuit current gain of the device which is an important quantity in transistor 



150 



THE TRANSISTOR AS A TWO PORT NETWORK 



CHAP. 6 



amplifying stages. The Thevenin and Norton theorems are important con- 
cepts and simplify the circuit analysis. Finally, the h parameter representa- 
tion leads to more simple algebraic expressions for the performance quantities. 
The h parameter representation will be used in the next chapter to analyze 
the CB, CE, and CC stages. 

It is found in practice, that the two branch equivalent circuit is no dis- 
advantage in the analysis of multistage amplifiers. It is true that by using 
z or y parameters a complete matrix may be set up to represent the perfor- 
mance quantities of such an amplifier, but such a matrix is tedious to solve 
and gives ample opportunities for human errors. In addition, it is shown in 
Chap. 7 that it is wise to design an amplifier stage by stage, starting with the 
output and working back to the input. For the calculation of the performance 
quantities of individual stages, h parameters are the most simple to use. 

6-7 Low Frequency Measurements on Transistors 

The d-c measuring circuit given previously in Fig. 6-4 enables the static 
characteristics to be drawn. To find the small-signal a-c parameters, a 
separate measuring circuit should be used since the z, y, or h parameters 
require certain specified input or output a-c conditions to be established. 



Low frequency 
generator 




Low frequency (r^\ 
generator V,-/ 



FIG. 6-14. (a) Circuit for measuring z 22 . (b) An attempt at 
measuring z u . 

For the measurement of z parameters, the defining equations (6-12) and 
(6-13) show that either the input or the output circuits must be open- 
circuited for alternating currents. Figure 6- 14(a) shows a simple circuit 
arranged to measure 

Yi 

1 2 /i = 



Zoo — 



SEC. 6-7 



LOW FREQUENCY MEASUREMENTS ON TRANSISTORS 



151 



in the CB configuration. The input is open-circuited by means of an induct- 
ance L. The reactance of the coil must be much greater than the input resist- 
ance of the transistor which, in practice, is only a few ohms. Even at low 
frequencies, this can readily be achieved. Figure 6-14(b) shows an attempted 
measurement of 

The output resistance of a transistor is usually about one megohm, and thus 
the impedance in series with the output should be 100 megohms or greater. 
A resistance of this magnitude cannot be employed because it would disturb 
the d-c conditions. Using a choke in the position shown in Fig. 6-1 4(b) is 
not satisfactory since the impedance of the choke cannot be made high 
enough to approximate to the open-circuit condition. For alloy-junction 
transistors, frequencies of less than 1000 cps must be used to eliminate 
reactive effects in the device, and so the open-circuit output condition cannot 
be achieved. 



i — VW 



Low frequency £\ Cboke 

generator \^s ' 



V, 



T^zlOOOfxf 



(a) 



^V cc 



Choke 




^ j Low frequency 
generator 



(b) 



FIG. 6-15. (a) Circuit for measuring h ie and h fe . (b) Circuit for 
measuring h oe and h re . 

Measurement of the y parameters requires a short-circuit a-c condition. 
In the output circuit, this can be obtained by shunting a large capacitor across 
the output terminals. In the input circuit, it is necessary to produce an a-c 
short-circuit having an impedance much less than the input resistance of the 
transistor. This condition is difficult to achieve since the input resistance is 
usually only a few ohms. 



52 



THE TRANSISTOR AS A TWO PORT NETWORK 



CHAP. 6 



Inspection of the definitions of the h parameters given in Eqs. (6-16) and 
(6-17) shows that measurement of all four h parameters is relatively easy. 
Determination of h xl and h 21 can be carried out when the output circuit is 
short-circuited to a-c and h 12 and h 22 can be found when the input is open- 
circuited. These two conditions are easy to reproduce, and so the h para- 
meter measurements are the ones most frequently taken. In many cases it is 
desirable to measure the CE hybrid parameters directly. This can be done by 
using the circuits of Fig. 6-15. Figure 6-1 5(a) may be used to determine h ie 
and h fe . Assuming R ± is of the order of one megohm and R 2 is a few ohms, 



hi„ - 



RiVi 



and 



h -Mi 

h,e ~ V*R,, 



*8 Y S^2 

In Fig. 6-1 5(b), the input circuit is open-circuited as far as a-c is con- 
cerned, and 

V 2 



h nP = 



RoV, 



and 



h -£ 

re Vs 



The relative positions of resistance R 2 , battery V cc , and generator V s , which 
are in series in the output circuit is determined by the type of supplies and the 
voltmeter available. It is often necessary to ground one side of the voltmeter 
terminals to avoid stray pickup. This can be accomplished by using a shielded 

TABLE 6-2. Relationship between the parameters. 



A y = ynyii - yizyzi 

A z = ZnZ 2 2 ~ ^12^21 

A h = huh 2 2 - hi 2 h 2 i 









In terms of 






Matrix 


y 


z 


h 












I 


yu 


yi2 


?22 ~Z\2 

A s A z 


1 
hi 


~/*12 
fill 


y 


^21 


yi2 


~~ Z21 Z\\ 

A 2 A 2 


h 2 i 
fin 


A h 
fin 




^22 

Ay 


-yi2 
Ay 


Zn Z12 


^22 


fl\2 
fl 22 


z 














-J>21 

A^ 


yu 

Ay 


z 2 i Z22 


-h 2X 

h 2 2 


fl 2 2 




1 


-yi2 


*L hi 


hn 


hi2 


h 


yn 


yu 


Z22 Z22 








y2i 


Ay 


-Z21 j_ 


^21 


h 22 




yn 


yu 


z 22 z 2 2 







PROBLEMS 153 

1 : 1 transformer with its primary connected across R 2 , and its secondary 
connected to the voltmeter. If an oscilloscope is used as the measuring in- 
strument, it may be advisable to ground one side of R 2 and use an audio 
generator and a power supply which can be operated above ground potential. 
Table 6-2 shows the relationship between the sets of parameters and 
can be used to convert between any one set of parameters and the others. 

BIBLIOGRAPHY 

Hurley, Richard B., Junction Transistor Electronics, New York: John Wiley 
& Sons, Inc., 1958 

Riddle, Robert L. and Marlin P. Ristenbatt, Transistor Physics and Circuits, 
Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1958 

Skilling, Hugh Hildreth, Electrical Engineering Circuits, New York: John 
Wiley & Sons, Inc., 1957 

PROBLEMS 

6-1 Verify the performance quantities in Table 6-1 for the common emitter 
stage using the T equivalent circuit. 

6-2 A CE stage has the following parameters: r b = 750 ohms, r e = 13 
ohms, r c = 1.5 megohms, r m = 1.47 megohms. If the resistance, R s , of 
the input generator is 1000 ohms and R L = 5000 ohms, draw an equiva- 
lent T circuit and find the current gain. Compare the result obtained 
with the value found by substitution in the appropriate equation in 
Table 6-1. 

ans. 42.0, 42.6 

6-3 What are the characteristics of a useful general purpose amplifier? 
Explain what is meant by "small-signal" conditions. What are the 
limitations placed on z n , z 12 , z 21 , and z 22 if the device is to be a linear 
current amplifier over a wide range? 

6-4 By circuit analysis, verify that Eqs. (6-12) and (6-18) apply to Fig. 6-7 
and that Eqs. (6-14) and (6-19) apply to Fig. 6-11. 

6-5 Derive the values of h lu h 12 , h 21 , and h 22 in terms of the z parameters 
and check your results from Table 6-2. 

6-6 Which internal equivalent impedances affect transistor operation at high 
frequencies? Identify these impedances in Fig. 6-12, and justify your 
answer. 

6-7 Why is it difficult to measure y and z parameters? Explain why it is 
easier to measure the hybrid parameters. 

6-8 Describe experiments to measure the h parameters giving the values of 
all circuit components used. 



7 



Low 

Frequency 

h Parameter 

Representation 



In this chapter, the h parameters are used in an 
analysis of the single-stage transistor amplifier. The 
performance quantities are calculated for the 
general representation, and then the CE, CB, and 
CC stages are compared. The discussion is limited 
to low frequencies, where the reactive effects of the 
transistor can be neglected. Curves of the variation 
in the h parameters of a typical transistor with 
emitter current and collector voltage are given. 



SEC. 7-1 



THE SMALL-SIGNAL h PARAMETERS 



155 



7-1 The Small-signal h Parameters 

In Sec. 6-2, the h parameters were defined by the two equations : 

V l = h ll I 1 + h l2 V 2 
h = hih + h 22 V 2 
input impedance, 
reverse voltage gain, 
forward current gain, 
output admittance. 



(7-1) 
(7-2) 



where h xl 
h 12 



! 22 



h xl and h 21 are measured under a-c short-circuit output conditions. h 12 and 
h 22 are measured under a-c open-circuit input conditions. The general small- 
signal two generator representation is shown in Fig. 7-1. 



h 



hu 



V, 



IT 

t 12 y 2 



^21 h 



FIG. 7-1. The small-signal h parameter equivalent circuit. 

Equations (7-1) and (7-2) and the small-signal equivalent circuit apply to 
all three connections, CB, CE, and CC. However, the values of the para- 
meters will be different for the three cases. To distinguish among them, the 
lettered subscripts shown in Table 7-1 are used. The first letter of the sub- 
script describes the term (input, reverse, forward, or output) and the second 
letter denotes the circuit connection (base, emitter, or collector). Figure 7-2 
shows the h parameter equivalent circuit for the three connections. 

In this chapter, we will use the numbered parameters for the analysis of 
the general equivalent circuit, and then substitute the appropriate lettered 
parameter for the actual circuit under consideration. 

TABLE 7-1. Numbered and lettered h parameters. 
Term 



Input resistance 
Reverse voltage gain 
Forward current gain 
Output admittance 



Numbered 


CB 


CE 


CC 


parameter 


parameter 


parameter 


parameter 


An 


hib 


hie 


h ic 


h X2 


h rb 


hre 


hrc 


fl 2 l 


h fb 


hfe 


hfc 


h 2 2 


hob 


hoe 


hoc 



56 



LOW FREQUENCY h PARAMETER REPRESENTATION CHAP. 7 



h ib 
O 1 1- 



Vi 



T 

h rb V 2 Q hftli 



(a) 



O 1 h 



v i h re V 2 Q) h f^ 



o 1 V 



v 2 @ 



(b) 



hfch 



(c) 



FIG. 7-2. /i parameter equivalent circuits: (a) CB circuit; (b) CE 
circuit; (c) CC circuit. 



7-2 Analysis of the General h Parameter Equivalent Circuit 

Figure 7-3 shows the h parameter equivalent circuit of the transistor with 
applied voltage sources and series resistances in the input and output circuits 
(see also Fig. 6-5). 

The five performance quantities below will be calculated at low fre- 
quencies. 



h 



An 



Vi A»Va@ 



hnh 




FIG. 7-3. General equivalent circuit of a transistor amplifying 
stage. 



SEC. 7-2 ANALYSIS OF THE GENERAL h PARAMETER EQUIVALENT CIRCUIT 



157 



Input resistance, R x = -= - 



Current gain, A t = 



Output resistance, R = -^ 

'2 



Voltage gain, A v = y 



Power gain, A p = A t A v 

For the input circuit of Fig. 7-3 

V x = hxh + h 12 V 2 
and for the output circuit 

v 2 h 22 + h 21 h - h - Fl ~ l K2 

or K L = // 2 i^lA + V 2 (l + /r 22 * L ) 



(7-1) 



(7-3) 



A. Input resistance. Solving Eqs. (7-1) and (7-3) for J x gives 





'.=i 


Vl 


h 12 
1 + h 22 R L 


where D = 


^11 *12 

/j 21 /? t 1 + h 22 R L 






= Ml + h 22 R L ) - h 12 h 21 R 


L 


= A h R L + h lx 




if A h = /in/Zaa - /Z 12 /?2i 




When F L = 0, 




, Kid 
1 J' 1 /? 


+ h 22 R L ) 

L + /*n 


and so 


R t = 


1 


A h R L + hi 



(7-4) 



B. Vo/tage gain. When K L = 0, from Eqs. (7-1) and (7-3) 



V * = D 



h 21 R L 



Thus, 



A -^ 



A h R L + An 

/*21*L 



A h R L + An 



(7-5) 



|58 LOW FREQUENCY h PARAMETER REPRESENTATION CHAP. 7 

C. Current gain. In the output circuit of Fig. 7-3 with V L = 0, 
Vo = -LR T = h " h * Jl 



since the current I 2 - h 2 Ji is flowing through the admittance h 22 . 
Hence, 



a-£- 



h 21 



h 1 + h 22 R l 



(7-6) 



D. Output resistance. In the input circuit of Fig. 7-3, when V s = 0, K x = 
-/i/? s , and so Eq. (7-1) can be rewritten as 

= hQi^ + R s ) + /* 12 K 2 (7-7) 

The output equation (7-2) for the two port network is unchanged and is 



h = h 21 A 4- h 22 V 2 



Hence, 



/*n + Rs 





h 2 i 


h 


/>n + Rs 


^12 


h 21 


^22 



/ 2 (^n + R s ) 



(hn + ^ s )/* 22 ~ h 12 h 21 



giving 

E. Power gain. 



R = Yl= h " + R s 



I 2 A h + h 22 R s 

a - a a - -HiRi 

P ~ v ' ~ (A k Rl + AiiXl + h 22 R L ) 



These results are summarized in Table 7-2. 

TABLE 7-2. Performance quantities for the h parameter equivalent circuit. 
Performance quantity Symbol Value 



Input resistance 


Ri 


Output resistance 


Ro 


Current gain 


A, 


Voltage gain 


A v 


Power gain 


A P 



A h R L + h lx 


1 + h 22 R L 


h\x + Rs 


A h + h 22 R s 


h 2 i 


1 + h 22 R L 


— h 2X R L 


A h R L + An 


-hhRi 



{A h R L + A n Xl + h 22 R L ) 
A h = huh 2 2 - h 12 h 21 



(7-2) 



(7-8) 



SEC. 7-2 



ANALYSIS OF THE GENERAL h PARAMETER EQUIVALENT CIRCUIT 



159 



Before the values of these quantities are determined for the three specific 
connections, the relationship of the parameters of the three circuits will be 
discussed. 

7-3 Relationship of the Three Sets of h Parameters 

It will be shown later that the CE circuit is the most used transistor amplifier. 
Therefore the CE parameters, h ie , h fe , h re , and h oe are of most interest. 
Transistor manufacturers vary widely as to which parameters they specify. 



h 



Y 



v, 



Z 



A 



h\] 



V 2 V, h 12 V 2 Q 



h 2x I x 



FIG. 7-4. CB equivalent circuit using numbered parameters. 

The current gain, h fe , will always be given since it is of great importance, but 
the other three may sometimes be omitted. A few years ago only CB para- 
meters were specified, and it was necessary to convert from one set of para- 
meters to the other. Figures 7-4 and 7-5 show how this conversion may be 



o- 



i; 



vi 



v; 



v; 



viQ 



hfe III 



v; 



FIG. 7-5. CE equivalent circuit using lettered parameters. 

effected. The CB circuit using numbered parameters is shown in Fig. 7-4. 
(Whenever numbered parameters are shown without further description, they 
refer to the CB connection.) Figure 7-5 gives the similar equivalent circuit 
for the CE connection where lettered parameters are used. The CB circuit 
may be arranged to give a CE connection as in Fig. 7-6. This last figure 
must correspond exactly with Fig. 7-5 where 

v 2 = v 2 - v[ 

and l x is the current flowing upwards through h lu that is, 

h = -d'i + Q 



60 



LOW FREQUENCY h PARAMETER REPRESENTATION CHAP. 7 



By means of this equivalence, it can be shown that 

h = /?11 

it ,p 



Ke = 

flnr = 



A h 


+ 


h 2 i - h 12 
& + h 21 


+ 


1 


A h 


+ 


1 + h 21 - 


■ h 


12 


A h 


+ 


hi - h 12 
h 2 2 


+ 


1 



I{ 




FIG. 7-6. The CB numbered h parameter equivalent circuit re- 
arranged as a CE connection. 

For a typical transistor, 

A h - h 12 « 1 + h 21 and h 21 » A h 

and these equations may be simplified to the values given in Table 7-3. The 
corresponding equations relating the CC h parameters to those of the CB 
connection are also given in this table. 

TABLE 7-3. h parameter equivalents of CE and CC circuits 
in terms of CB parameter (approximate values). 



CB 

parameter 
numbered lettered 



CE 

parameter 



hi2 

#22 



hrb 

hfb 

hob 



hie — 
hre = 
hfe = 

hoe = 



1 + h 21 
A h - h 12 

1 + ^21 

— h 2 i 

1 + ^21 

h 2 2 
1 + h 21 



CC 

parameter 


hu 


An 


1 + h 21 


h rc 


= 1 


hf C 


-1 


7 1 + h 2 i 


hnr 


' h 22 



1 + h 21 



SEC. 7- 



COMPARISON OF THE THREE CONFIGURATIONS 



161 



7-4 Comparison of the Three Configurations 

Three of the performance quantities which determine the properties of the 
amplifying stage have been given as a log-log plot in Figs. 7-7, 7-8, and 7-9. 
Curves are shown for all three connections using a typical transistor having 
h parameters values as given in Table 7-4. 

TABLE 7-4. Typical h parameter values. 



Parameter 


CB value 


CE value 


CC ya/we 


ftii 


h ib = 40 ohms 


h ie = 2000 ohms 


h ic = 2000 ohms 


hi-2. 


h ro = 4 x 10~ 4 


h re = 16 x 10- 4 


Arc^ 1 


^21 


h fb = -0.98 


h fe = 49 


h fc = -50 


^22 


hob = 10" 6 mhos 


h oe = 5 x 10" 5 mhos 


h oc = 5 x 10~ 5 mhos 


A H 


A hb = 432 x 10- 6 


A he = 0.0218 


J hc = 50.1 



A. Common base circuit. In Fig. 7-7, the current gain A { is plotted as a func- 
tion of the load resistance R L over the wide range, R L = 100 ohms to 



100 



























CB 



10 2 10 3 10 4 10 5 10 6 

Load resistance R L in ohms 

FIG. 7-7. Variation in current gain with load resistance. 

1 megohm. For the CB stage, A { has the maximum value of 0.98 (magnitudes 
only are shown on the curve), and this value is maintained as the resistance is 
increased, up to load resistances on the order of 10 5 ohms. However, a 
current gain of less than unity makes this stage unsuitable for use in a multi- 
stage current amplifier. 

Figures 7-8 and 7-9 show that the input resistance of the CB stage is on 
the order of 100 ohms and that the output resistance is above 10 5 ohms. The 
CB stage may therefore be used as a buffer stage having a low input resistance 
and a high output resistance. The feedback term from output to input, h rb , is 
the lowest value for the three circuits so good isolation between output and 



162 



LOW FREQUENCY h PARAMETER REPRESENTATION 



CHAP. 7 



input can be achieved. This stage does not introduce phase reversal, and, as 
will be shown later, it has a wider frequency response than the other two 
connections. 

B. Common collector circuit. Figure 7-7 shows that it is possible to achieve a 
high current gain with one CC stage. However, a high current gain per stage 
cannot be achieved if CC stages are cascaded to form a multistage amplifier. 
The reason for this can be seen upon examination of Fig. 7-8. By comparing 
the dotted line representing R t = R L and the curve of input resistance of the 
CC stage we see that the input resistance is higher than R L by at least one 
order of magnitude up to R L = 10 5 ohms. Thus for identical, cascaded CC 



io 6 



io 5 



10" 



05 

y 

a 

CQ 

W 

1 103 







^CC 


/ 
/ 
/ 
/ 
/ 
/ 






/ 
/ 
/ 

/ 






/ 
/ 
/ 








/ 


/ 
/ 

/ 
/ 
/ 
/ 

/ 
/ 


















CB 





10 2 



10 2 IO 3 IO 4 IO 5 IO 6 

Load resistance Rl in ohms 

FIG. 7-8. Variation in input resistance with load resistance. 



stages, the load resistance of a stage will bypass the input resistance of the 
next stage to such an extent that less than 10 per cent of the current will pass 
into the succeeding stage. This reduces the overall current gain of the stage 
to a value close to unity and makes the cascaded stages of little use. 

One feature of the CC circuit is that the reverse feedback term, h r0 , is close 
to unity. This means that the output and input voltages will be approxi- 
mately equal. Thus, however the input alternating voltage from base to 
collector alters, the output voltage from emitter to collector will follow. For 
this reason, the CC circuit is often known as the emitter follower. Although 
this circuit is unsuitable for cascading in a multistage amplifier, it does find a 



SEC. 7-4 



COMPARISON OF THE THREE CONFIGURATIONS 



163 



use where its high input resistance and unity voltage gain characteristic can 
be utilized. 

C. Common emitter circuit. The CE stage is seen to have a high current gain 
and input and output resistances which are intermediate between the values 
found for the CB and CC stage. When used in a cascaded amplifier, the out- 
put resistance of the CE stage given here is between 10 4 and 10 5 ohms, and 
the input resistance is about 2000 ohms. From Fig. 7-7, the current gain per 
stage in a cascaded amplifier where the input resistance of one stage forms the 
load of the previous stage approaches the short-circuit a-c value, h fe . Unlike 
the CC stage, the load resistance can be made higher than the succeeding in- 
put resistance (2000 ohms), and so almost all of the alternating current leaving 
the stage enters the base lead of the following transistor. 

io 6 



105 



10 3 



10 2 



10 





CB 










_ CE 




















y^CC 




~~" 









10 5 



10 f 



10 3 10 4 10 5 

Source resistance R s in ohms 

FIG. 7-9. Variation in output resistance with source resistance. 

The CE stage introduces a phase reversal of 1 80° between output and input 
currents similar to a vacuum tube operated in the conventional manner. The 
frequency response of the CE stage is not as high as the CB stage, but, by 
selection of a suitable transistor, multistage amplifiers having a response up 
to hundreds of megacycles may be obtained. 

7-5 Variation in h Parameters with U and Vce 

In general, the values of the h parameters are dependent on the quiescent 
operating point of the transistor. It is difficult to predict these variations with 
accuracy since there are many factors to take into account. Figures 7-10 and 



164 



LOW FREQUENCY b PARAMETER REPRESENTATION CHAP. 7 



a II 

TO K«h 

s 



1.0 



0.6 



0.2 



- 










A oe 


- 






















/C 


- 










hf e 


^ 










h ie 


- 












- 













0.2 



1.0 

7 £ in milliamps 



FIG. 7-10. An example of the variation in CE parameters with 
emitter current. 



7-1 1 show h ie , h oe , h re , and h fe for an alloy-junction transistor as a function of 
I E and V CE . These curves were drawn from data supplied by the manu- 
facturer. In Fig. 7-10, h fe is shown to increase with I E . For values of I E 
higher than shown in this graph, h fe would reach a maximum and decrease 



3.0 



2.0 



O ;£ 

s g 



as 



1.5 



- 








s^*„ 






^reX/ 


h oe ^^^^ 






hfe Jy n oe 


- ;gg 










] 


i i i 





volts — *• 



FIG. 7-1 1. An example of the variation in CE parameters with 
collector voltage. 



SEC. 7-5 VARIATION IN h PARAMETERS WITH / £ AND V CE 165 

(see Sec. 9-7). In Fig. 7-11, h ie and h fe steadily increase with V CE , whereas 
h re and h oe have a minimum value at about V CE = 8 volts. It is usually 
unwise to attempt to extrapolate curves of this nature much beyond the stated 
limits. 

There are wide variations in the values of the parameters of transistors 
which have the same type number. There are also substantial variations in 
the form and the relative positions of the curves shown in Figs. 7-10 and 
7-11. Manufacturers commonly specify only the average values of the h 
parameters at one operating point. If it is necessary to know the exact value 
of a particular transistor parameter, a measurement should be taken under 
the desired conditions. 

BIBLIOGRAPHY 

Fitchen, Franklin C, Transistor Circuit Analysis and Design, Princeton, 
N.J. : D. Van Nostrand Company, Inc., 1960 

Joyce, Maurice V., and Kenneth K. Clarke, Transistor Circuit Analysis, 
Reading, Mass.: Addison- Wesley Publishing Company, Inc., 1961 

Riddle, Robert L., and Marlin P. Ristenbatt, Transistor Physics and Circuits, 
Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1958 

PROBLEMS 

7-1 A transistor has the following parameters: h xl = 45 ohms, h 12 = 4 
x 10~ 4 , h 21 = -0.98, h 22 = 10" 5 mhos. From Table 7-3, find the 
parameters of the CE and CC circuits. 

7-2 Using Fig. 7-2(c) as the equivalent circuit for the CC connection with a 
source resistance R s and a load resistance R L , write down the circuit 
equations and solve for the power gain. 

7-3 Explain the advantages and disadvantages of using h parameters for 
transistor specification. 

7-4 The transistor of Prob. 7-1 is operated in the CE connection with a load 
resistance R L and with a resistance R A between the collector and the 
base terminals. Find the current gain of the circuit. 

7-5 Find the input resistance, the output resistance, and the current gain 
for a CE circuit with R s = 1000 ohms and R L = 5000 ohms. The CB 
specification of the transistor is h ib — 40 ohms, h Th = 4 x 10 ~ 4 , 
h fb = -0.97, h ob = 10~ 6 mhos. 

7-6 A crystal microphone requires a load resistance of about one megohm 
and the output from the microphone is to be fed into a CE amplifying 
stage of input resistance 2000 ohms. Design a "buffer" stage to go 



166 LOW FREQUENCY h PARAMETER REPRESENTATION CHAP. 7 

between the microphone and the amplifier. Use a transistor with 
specifications given in Table 7-4 and explain the reasons for your choice 
of circuit. 

7-7 A transistor is operated as a CE amplifier with a resistance in the emitter 
lead. If the parameters of the transistor are h iei h re , h fe , and h oe , find the 
h parameters of a two port network which includes the transistor and 
the resistance. 

7-8 Two CE stages are coupled in series so that all of the alternating current 
from the collector of the first transistor flows into the base lead of the 
second transistor. If the two transistors are identical and have h 
parameters h ie , h re , h fe , and /? oe , what are the h parameters of the two 
port network which includes both transistors? 

7-9 Explain why CB or CC stages are never cascaded in a multistage 
amplifier. What are the chief features of a CE multistage amplifier ? 

7- 1 Show that for the CE configuration : 

"ie = r b ~^~ ~i ' "re ~ 



1 - « ' r„ - r„ 

1 



n fe — t _ ' n oe a 

i a 'c 'm 

7-11 A transistor having h parameters as given in Table 7-4 is used in a CC 
circuit. If the load resistance is 10,000 ohms and the source resistance 
is 1 megohm, calculate R h R , A u A v , and A p . 



8 



Single 

and Multistage 

a-c Amplifiers 



The CE transistor circuit, although not perfect, has 
considerable merit as an amplifier. It has a rela- 
tively high current gain, it is small and light, it 
requires low voltage supplies only, and it has a high 
efficiency. There are two undesirable features, 
however, which must be taken into account. The 
more important of these is the variation which is 
exhibited by almost all transistor characteristics and 
parameters as the temperature of the device 
changes. A second adverse property of present-day 
transistors is the large variation which exists 



168 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

between characteristics of different units of the same type number. These two 
factors will be shown to determine the circuits used for obtaining the d-c bias 
to the transistor. 

The operation of the transistor as an a-c amplifier in the manner indicated 
in the two preceding chapters is conditional upon obtaining and main- 
taining a suitable quiescent operating point. Thus it is desirable to have the 
transistor operating with the same a-c gain, and with the same direct current 
and voltage in its output circuit, even though the input and output transistor 
characteristics depend upon temperature. In this chapter, it is shown that 
operating point stability can be achieved by careful design of the d-c biasing 
circuit. We will limit the analysis to the two most-used circuits at low 
frequencies: (a) a small-signal CE amplifier where current gain is of the 
greatest importance, and (b) a CE amplifier which is designed for maximum 
power output and where current gain is of secondary importance. 



8-1 The Characteristic Curves 

Figure 8-1 shows the direct currents which flow in the three transistor leads. 
The transistor is assumed to be in the CE connection. From Chap. 5 it has 

been shown that the transistor satisfies the 
i c equations : 





*C = *E ~ *B 


(8-1) 




'c — a *E "1" *CBO 


(8-2) 


t'. 


and by eliminating I E , 




i directions 


t a T , Icbo 

J C — i I B 1 i 

1 — a 1 — a 


(8-3) 



of current. 

where I CBO is the reverse-biased collector- 
junction current which flows when the emitter current is zero. We may 
write 

(8-4) 
(8-5) 



(8-6) 



If it is assumed that the quantity a is independent of I c and that I CEO can 
be neglected, Eq. (8-6) indicates that the collector current of a transistor is £ 







P - 1 - a 






1 -B 




l-« P 


when Eq. (8-3) becomes 








Ic 


= rIb + Iceo 


where j 


*CEO 


= (P+ WCBO 



SEC. 



THE CHARACTERISTIC CURVES 



69 



times the base current. In the small-signal two port network theory de- 
veloped in Chaps. 6 and 7, it was shown that the small-signal short-circuit 
current gain of the transistor in the CE connection was h fe . Hence, j8 « h /e . 
However, h fe is a function of I c , V CB , and temperature, and Eq. (8-6) relates 
to the d-c operation of the transistor so that j8 in this equation cannot simply 
be replaced by the small-signal value h fe . Therefore, we will use the symbol 
h FE (using capital subscripts) to denote the d-c or large signal value of j8, where 
it is necessary to distinguish between the a-c and the d-c case. Similarly, h fb 




FIG. 8-2. Output characteristics of a typical pnp germanium 
alloy junction transistor. 



which has been defined previously as the short-circuit small-signal current 
gain in the CB connection will be replaced by h FB for d-c operation. The 
symbol a wilt be retained when no confusion will arise. Both a and p are 
terms in common usage in transistor circuitry. 

The output characteristic curves of Fig. 8-2 can be explained by reference 
to Eq. (8-6). The curve labeled I B = 0, represents the current I CEO , sl current 
j8 + 1 times I CBO , If j8 and I CBO were constant in Eq. (8-6) all the curves 
would be equally spaced, straight lines parallel to the V CE axis. At low values 
of Vce, such that the collector-to-base junction is just biased in the forward 
direction, the current falls to zero and transistor action ceases. In this region, 
Eq. (8-6) cannot be applied. Over the rest of the range, j8 and I CBO are seen 
to be dependent on V CE and I c . The value of j8 (h FE ) can be calculated from 
the figure to be Al c jAI B to give an average value between the points A and B. 

There is little significant difference between the output characteristic of a 
silicon and a germanium transistor. For silicon, the value of I CBO is much 



170 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 8 



smaller but the form of the curves is similar. The input characteristics of 
germanium and silicon transistors are different, however, as shown in Fig. 8-3. 
For both diagrams, the base current reverses for V BE « 0. This is due to the 
flow of current in the base lead (~ —I C bo) when the collector to base is re- 
verse biased. The input voltage required to produce I B > is considerably 



200 




10 volts 



700 


/ 




600 


/ 




m 50 ° 

CO 

3. 


/ 




.2 400 


/ 




^ 


I 




300 


\ 




200 


Jv CE = 


- 10 volts 


100 


y 





(a) 



•0.1 -0.2 

V BE in volts 



(b) 



■0.2 -0.4 -0.6 -0.8 -1.0 
V BE in volts 



FIG. 8-3. Input characteristic of pnp alloy junction transistors: 
(a) germanium; (b) silicon. 



different in the two types of transistors. It is of the order of 150 millivolts for 
germanium and 500 millivolts for silicon devices. The input curves lose their 
exponential form and become approximately straight lines at high values of 
base current. This occurs when the current is so high that most of the base- 
to-emitter voltage drop takes place across the ohmic resistance of the base. 

8-2 The Load Line 



Figure 8-4 shows the circuit of a CE amplifying stage. The supply battery 
V cc is in series with the load resistance R L and the collector-to-emitter 

terminals. This is similar to the output 
ic circuit of nearly all electronic devices. 

As far as alternating currents are con- 
cerned, the resistance of the supply can 
be ignored, and R L is the load into 
which the transistor is working. I c is 
dependent upon the supply voltage F cc , 
the load R L and the collector-to- 
emitter voltage V CE as given in Eq. 
FIG. 8-4. The simple CE circuit. (8-7). If we designate all the voltages in 




SEC. 8-2 



THE LOAD LINE 



171 



the conventional manner with the terminal indicated by the first subscript as 
positive, we can write 



V cc = -I C R L + V { 



CE 



(8-7) 



where the supply voltage, V cc , and the voltage, V CE , are negative values for 
the pnp transistor. Equation (8-7) can be written in the form 



Ir = 



CE 



R, 



Vcc 
Rl 



(8-8) 



and is plotted in Fig. 8-5 where, by convention for a pnp transistor, the 
negative values of V CE are plotted. On these axes, Eq. (8-8) is the equation of 
a straight line of negative slope (l/R L ), of intercept on the I c axis of — V CC /R L , 



600 p a 




500 pa 



400 pa 



300 pa 

inusoidal variation 
in I n 

200 pa 



100 pa 



7 n = 



V CE in volts 
FIG. 8-5. The load line plotted on the output characteristic. 

and which intercepts the — V CE axis at V cc . The output characteristics shown 
are those of a medium power alloy-junction transistor with V cc = — 8 volts 
and R L = 160 ohms. The straight line representing Eq. (8-8) is known as the 
load line, and its intersection with the output characteristics gives a set of 
quiescent operating points (Q points) for the device. Points Q , Q u Q 2 , 
Q 3 , . . . are obtained as I B is changed. In the case shown in Fig. 8-5, the 
corresponding values of I B are 0, 100, 200, 300 /xa respectively. At all of the 
Q points, the circuit equation (8-8) is obeyed, and the device is operating on 
one of the characteristic curves shown in the diagram. 

In Fig. 8-5, I B = 200 /xa has been fixed to give a Q point at Q 2 . Then 
a sinusoidal variation of amplitude 100 /xa has been impressed upon I B . 
During this time, the circuit must still obey Eq. (8-8), and so the operating 



172 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

point moves from Q 2 up to Q 3 down to Q 1 and back to Q 2 remaining on the 
load line all the time. The output current I c as a function of time is shown 
to the left of the characteristic curves. 

The selection of the Q point is decided upon by reference to the purpose 
of the stage. For a small-signal amplifier where power is to be conserved, a 
Q point would be chosen so that the quiescent value of I c is as low as possible, 
provided that the output remains a faithful reproduction of the input signal. 
For an amplifier which is required to deliver a substantial amount of power 
into a load, the Q point should be selected so that the quiescent current 
is about one half of the maximum permissible collector current of the tran- 
sistor. In this case, the allowable current swing above and below the Q point 
before distortion occurs is at a maximum (see Sec. 8-5). 

The analysis which has been given in this section may be applied to the 
output circuit of many electronic devices. In the case of a transistor, how- 
ever, the establishment of the quiescent operating point by fixing the base 
current results in a circuit having poor temperature stability. The reason for 
this is explained in the next section. 

8-3 Operating Point Stability 

It has already been shown in Part 1 of this book that many properties of 
semiconducting materials are dependent upon temperature. A transistor is 
usually operated so that its temperature is above that of its surroundings. 
The thermal capacity of the device is low because it is small and light. Thus 
the transistor will soon reach its operating temperature which is a function of 
its power dissipation, its thermal capacity, and the ambient conditions. Since 
many transistor parameters and characteristics are markedly temperature 
sensitive, the quiescent operating point should be established with the purpose 
of stabilizing the gain and output of the stage. The methods of achieving 
such a design will be explained. It will also be shown that, by poor design, 
the transistor can be made to heat itself up to such a temperature that it is 
destroyed. This process is known as thermal runaway and can be carried to 
completion only once for each transistor. 

An increase in temperature has an important effect on three transistor 
quantities: I CBO , a, and V BE . In addition, there are likely to be significant 
differences in the measured values of I CBO and a between transistors of the 
same type. The spread in the values of these quantities is troublesome in 
circuit design, and the manufacturers are continually trying to reduce it. It 
will be shown that circuits can be devised that will reduce the effect of changes 
in I CBO or a whether these are the result of changes in temperature or of the 
replacement of the transistor by another of the same type. 

The typical variation of I CBO with temperature is given in Fig. 8-6 for 
silicon and germanium transistors. A considerable spread will be found in the 
values of I CBO and in the slope of the curves when measurements are taken on 



SEC. 8-3 



OPERATING POINT STABILITY 



173 



1000 



100 



.9 
o 

4> 



10" 



10 

























Ge 


rmanium^ 




















Silicon 







10 



20 30 

Temperature, °C 



40 



50 



FIG. 8-6. Value of I C bo for silicon and germanium transistors. 

several types of transistors. This is because surface effects in the transistor 
and leakage currents across the junction contribute to I CB0 and make it 
dependent upon the manufacturing techniques. Transistors made of silicon 
have a much smaller value of I CBO than those made of germanium because of 
the larger energy gap between the valence and conduction bands of the former 
material. In many cases, I CBO for silicon transistors can be neglected without 
appreciable error being incurred even for high temperatures, and then the 



1.2 
1.0 

0.8 











































20 30 

Temperature, °C 



40 



FIG. 8-7. Current gain (h FE ) as a function of temperature 
(normalized to 25° C). 



174 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

variation in a or V BE becomes of paramount importance. Measurements on 
both silicon and germanium transistors show that an increase in temperature 
of about 10° C results in I CBO being doubled, whatever its initial value. 

In a CE circuit, the quantity h FE (the d-c value of ]8) determines the direct 
current gain. Furthermore, j8 [= a/(l — a)] is more sensitive to changes in 
temperature than is a, and so the temperature variation in j8 is more likely to be 
given by the manufacturers. In Fig. 8-7, this value has been normalized to 
25° C and is given only over the temperature range 0° to 50° C. Maximum 
and minimum values of h FE (or possibly h fe ) for a given transistor at 25° C will 
usually be given in the data sheet supplied by the manufacturers. Thus a 
typical transistor may have the following specifications: 



h FE at 25° C ' 
I c = 1 ma 
V CE = -lv . 



> 20 (minimum) 35 (typical) 60 (maximum) 



The base-to-emitter voltage necessary to produce a given emitter current 
for a germanium or a silicon transistor exhibits a temperature coefficient, and 
the voltage decreases at the rate of about 2.5 millivolts per degree centigrade. 
This is predictable from simple theory in the case of germanium, but the 
theory predicts a somewhat higher value for silicon which is not found in 
practice. There is little variation in this temperature coefficient between 
different transistors. For pnp transistor operation, the base-to-emitter voltage 
is negative. Thus, as the temperature rises, V BE becomes less negative, and so 
the temperature coefficient is a positive quantity. 

From Figs. 8-6 and 8-7 and from the information in the last paragraph, 
we may summarize the variation in 7 CBO , h FE , and V BE over the temperature 
range 0° to 50° C in the following three equations. 



AIcBO C\C\1T 



ampsT" 1 (8-9) 



T°C 



Equation (8-9) states that there is a change in I CBO of the order of 7 per cent 
per °C of its value at the temperature under consideration. 



^ ~ 0M2h FE 



C 1 (8-10) 



25°C 



Equation (8-10) states that there is a change in h FE of the order of 1.2 per 
cent per °C of its value at 25° C. 

^P ~ 0.0025 volts °C" 1 (8-11) 

Equation (8-11) states that, for a. pnp transistor, V BE required for a constant 
emitter current becomes more positive at the rate of 2.5 mv per °C. 

Note the difference in the temperature coefficients of I CBO and h FE . I C bo 



SEC. 



OPERATING POINT STABILITY 



175 



shows an exponential dependence upon temperature, and thus its temperature 
coefficient increases as the temperature rises. The temperature coefficient of 
h FE is constant, and thus h FE at any temperature may be referred to its 25° C 
value. For the temperature range 25° to 35° C, I CBO approximately doubles 
(i.e., shows an increase of 100 per cent), and h FE increases by 12 per cent. It 
will be concluded, therefore, that as the temperature increases, I CBO increases 
more and more rapidly with respect to the increase in h FE as the Table 8-1 
shows. These values are only approximate, but they apply to many small 
germanium and silicon alloy junction transistors. 

TABLE 8-1. Variation in I CB o, h FE , and I B normalized to 
25°C and their typical values for a fixed-base biased tran- 
sistor circuit. 





Normalized to 25° C 


Typical value for 
germanium at 

25° C 


Typical value 
for silicon at 

25° C 


Temperature 


25° C 


35° C 


45° C 


IcBO 

h F E 

h 


1 
1 
1 


2 

1.12 

1.025 


4 

1.24 

1.05 


1 /xa 
50 
100 /^a 


0.01 ^a 
50 
100 n& 



In order to estimate the effects of temperature on the operation of the 
transistor, Eq. (8-12) will be used to represent the output characteristics in 
the CE connection. Here h FE is used in place of j3 of Eq. (8-6). 



h = h FE I B + (h FE + l)/ c 



(8-12) 



As a simplification, we shall assume an "idealized" transistor where I CBCh 
h FE , and I B are independent of I c and V CE . The output characteristics of this 
transistor are straight lines parallel to the V CE axis and are shown in 
Fig. 8-8(a) for room temperature. In this diagram, the value of(h FE + 1)I C bo 
has been exaggerated for clarity. A load line has been drawn on the 



c 












\ 






\« 








hFE 1 !), 




, . , v . . 



(h FE + l)I CBO 

'IcEO 







\q 




h FE lB x 

1 


V 



(h' FE + 1)I'cbo 

= IcEO 



(a) 



(b) 



FIG. 8-8. Output characteristics of idealized transistor showing 
the load line (a) at room temperature; (b) at a higher temperature. 



176 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 



characteristics under the assumption that the transistor is connected in the 
circuit of Fig. 8-9. Now suppose that the temperature is increased so that 
the new currents and parameters are I' c , I' B , h' FE , etc., related by the equation 



Ic — h FE I B + {hpE + l)-'CBO 



(8-13) 



The output characteristics at the new temperature are given in Fig. 8-8(b). 
Here (h FE + \)I' C bo is considerably greater than its previous value, and also 
the spacing between the I' B1 , I' B2 curves has increased. As a result, the Q 
point shifts from Q to Q '. 

The circuit of Fig. 8-9 is known as fixed-base biasing. If we designate all 



R B 
r^VW 



v DD ^ 



Rl 

A/W-i 



^v, 



CC 



FIG. 8-9. The simplest d-c biasing circuit. 

the voltages so that the terminal indicated by the first subscript is positive, we 
may write the following equations. 

Vbb - Vbe 



In = 



(8-14) 



and 
Thus, 



/; = 



Vbb ~ V'e 

R„ 



Ah 
AT 



I'b 



AT 



Vbe ~ Ve 
ATR B 



^ Vbe 
ATR t 



i™^ 1 



(8-15) 



(8-16) 



where R B is in ohms, since AV BE /AT = 2.5 mv per °C for a pnp transistor 
from Eq. (8-11). Thus I B is also increased at the rate of 2.5/R B ma per °C; 
an effect which still further increases I c . 

Under normal operating conditions, a rise in temperature of 20° to 
40° C is common. I C bo> therefore, increases to many times its initial value as 
the circuit warms up and increases in h FE and I B also occur. The shift in 
Q point shown in Fig. 8-8 usually makes this fixed-base biasing circuit un- 
acceptable. 

For the simple fixed-base biasing circuit of Fig. 8-9, variations in I CBO , 
h FE , and V BE all contribute to an increase in I c with temperature. Other 
circuits will be given which reduce this dependence of I c upon temperature. 
Since I CBO is the quantity which shows the greatest variation, the differential 



SEC. 8-3 OPERATING POINT STABILITY 177 

coefficient dI c l8I CBO is a useful measure of the temperature stability of a 
circuit. Defining this quantity as 

Stability factor, S = ~^- (8-17) 

we can avoid taking temperature measurements and yet have an indication of 
how the circuit will behave in practice. With this definition, a high value of S 
indicates poor stability and a low value of S shows good stability. The lowest 
value of S that can be obtained is unity since I c must include I CBO , 

The stability factor of the circuit of Fig. 8-9 can be simply obtained since 
from Eq. (8-12), 

S = -^- = h FE +\ (8-18) 

al CBO 

if I B and h FE are regarded as independent of I cbo . This value of S is the 
highest that can be obtained showing that the circuit has very poor tem- 
perature stability. 

The stability factor S gives a measure of the change in I c with respect to 
I CBO whether the change is the result of temperature variation or due to the 
transistor being replaced. For example, consider transistor A having 
Icbo = 1 j^a at 30° C and h FE = 49 operating in a fixed-base bias circuit with 
the collector current I c = 0.5 ma. In this circuit, 

I ceo = {h FE + 1)1 cbo = 50 ^ a 
If the transistor were replaced by transistor B of the same type having the 
same value of h FE but with I CBO = 5 /xa at the same temperature, I CEO would 
rise to 250 t*a and increase I c by about 40 per cent. Changes as great as, or 
greater than, this are observed in practice for the fixed-base bias circuit 
where S = h FE + 1. The performance of the stage in this respect is bad. 

We have shown that I CBO varies more widely with temperature than any 
other quantity, and so stabilizing with respect to I CBO will adequately stabilize 
against temperature variations in h FE . However, h FE may be subject to varia- 
tion irrespective of changes in I CBO , such as when a transistor is replaced. We 
will use the symbol S h to indicate this variation in I c where, 

5 * - £ (8 ~ 19) 

For the fixed-base bias circuit of Fig. 8-9, 

1q = n FE I B + \h FE + 1)1 cbo 



Therefore, 



Su = Tsr- = 1b + I 



CBO 



die 

dh FE 

Ic ~ (h FE + l)/ CBO + h FE I c 
h FE 

Ic ~ Icb o 
h FE 



178 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 8 



Considering small changes AI C and Ah FE in I c and h FE respectively, we may 
write 



or 



Ah 

Ah FE 
Ale 

In 



[cBo\ 

ic! 



h FE \ 
h FE \ Ic J 



(8-20) 



Equation (8-20) is a useful way of stating the variation in I c as a result of 
changes in h FE . It shows that for the fixed-base bias circuit, where I CBO « I c , 
the percentage change in I c is the same as the percentage change in h FE . 
Since the variation in h FE between units may be 4- 100 per cent to —50 per 
cent, this circuit is not acceptable. 

8-4 Practical Biasing Circuits 

The most commonly used CE biasing circuit is that of Fig. 8-10. In this 
circuit, a resistance R E and a capacitance C E are in parallel in the emitter lead. 
For frequencies much higher than given by 

i _ R 

the reactance of the capacitance is much smaller than the resistance R E and 
may be neglected in practice. Thus for high frequencies, the stage reduces to 
the circuit of Fig. 8-9. 

For direct currents, the resistance R E forms a link between the input and 
output circuits since both I B and I c flow through this resistance to form the 



\AAn 



a-c o 



input signal 



V^ 



V„r,-=r 



Rrr T C 



'cc 



FIG. 8-10. CE biasing circuit with a resistance in the emitter lead 



current I E . Furthermore, the voltage developed across R E by the output cur- 
rent opposes the input voltage from the battery supply V BB . This is known as 
current negative feedback because the reverse voltage developed in the input 
circuit is proportional to the output current. Voltage negative feedback (see 



SEC. 8-4 PRACTICAL BIASING CIRCUITS 179 

Fig. 8-13) may be produced by feeding into the input circuit a proportion of 
the output voltage in such a way that it opposes the input voltage. The 
general properties of feedback circuits will not be discussed here but may be 
referred to in any general textbook on electronics. We will show, however, 
that the value of the stability factor can be reduced by this technique. 

The following equations apply to the circuit of Fig. 8-10. For the 
transistor, 

Ic = h- h (8-1) 

h = h FE I B + (h FE + \)I CBO (8-12) 

By mesh analysis of the base-to-emitter circuit, 

Vbb = -IeRe - IbRb + Vbe (8-21) 

Substituting for I E and I B from Eqs. (8-1) and (8-12), Eq. (8-21) becomes 

Vbe ~ V BB = I c [r e + ^J^] - [^J~]Icbo(Rb + Re) (8-22) 

If the variations in V BE and h FE are ignored, Eq. (8-22) may be differentiated 
with respect to I CBO to find the stability factor. 

Therefore, 

s = (h FE + l)(R B + R E ) = R B + R E = 1 + R B /R E (8 _ 23) 
(h FE + l)R E + R B „ R B . R B 



h FE + 1 (h FE + I) Re 



When R E -+ 0, S -> h FE + 1 
When tf R -^0, 5->l 



When R E is zero, the circuit becomes identical with the previous circuit 
given in Fig. 8-9. When R B is zero and R E is finite, this circuit becomes a com- 
mon base configuration as far as direct currents are concerned. Assuming 
V BE is constant, I E is fixed by V BB and R E and is not temperature dependent. 
Hence I c can only increase as much as I CBO increases. This results in S = 1. 
The circuit remains a CE amplifier for alternating currents since the signal is 
fed to the base terminal, and C E bypasses R E at signal frequency. 

In practice, a compromise between the two extreme values of S is chosen. 
If R E is made very high, it will dissipate too much power from the collector 
supply and will create an excessive voltage drop between the emitter and the 
common lead. If R B is made too low it will act as a shunt to the input signal 
current and reduce the effective gain of the stage. For low power stages, 
where the rise in temperature is small, S = 10 can often be tolerated. For a 
typical value of h FE = 50, this requires that R B jR E ^11 from Eq. (8-23). 



80 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 



*CO- 1(- 



input signal 



Ri 




■^V r 



^c, 



FIG. 8-1 I. A practical biasing circuit. 

The variation in I c because of changes in h FE will now be investigated. 
Differentiating Eq. (8-22) with respect to h FE gives 



= S h R E + (R e + * B )( h "% E T ° ) - I. 
Thus, 



(R E + R B ) r , , 

CBO T2 [HFE ~ {"FE + UJ 

rlpv 



e = _^ = (^ ~ W Cgg + gg) = (Jc ~ Jcbo) S 

h 8h FE h FE h FE R E + R E + R B h FE (h FE + 1) 



Considering small changes in I c and h } 



Ah 

Ic 



Ah, 



(ft 



FE 



I /, _IcBo\ 

+ 1) V Ic J 



(8-24) 



This represents an improvement by the factor S/(h FE + 1) over the perform- 
ance of the fixed-base bias circuit of Fig. 8-9 (Eq. 8-20). 

To avoid having two voltage supplies, the base current can be obtained 
from the collector supply V cc . Figure 8-1 1 shows a practical circuit which is 
commonly used to achieve this. Resistances R 1 and R 2 form a voltage 
divider across the collector supply and can be replaced by the Thevenin 
equivalent consisting of a battery of voltage V CC R 1 I(R 1 + R 2 ) and a series 
resistance of R 1 R 2 /(R 1 + ^ 2 ) as shown in Fig. 8-12. With reference to the 
values used previously in this section, 

VccRi 



V„ n = 



Ri + Ri 



Rx 



ft, -L 



R1R2 
R1+R2 
AA/V 



CC R x + R 2 



FIG. 8-12. Two circuits which are equivalent. 



SEC. 8-4 



PRACTICAL BIASING CIRCUITS 



181 



and 



R* = 



R,R, 



Ri + R 2 

The circuit shown in Fig. 8-13 employs both current and voltage feedback. 
As before, a resistance is placed in the emitter lead to give current feedback 
under d-c conditions. The resistance R F feeds back output voltage from the 
collector circuit to the base and is responsible for negative voltage feedback. 
Writing the d-c equation for the circuit, we have 

V cc = -Ie(Re + Rl) - IbRf + Vbe (8-25) 

Substituting for I E and I B in terms of I c and I CBO from Eqs. (8-1) and (8-12), 



I C Bo{h FE + lXgg + Rl + Rf) 
R F + (R E + R L )(h FE + 1) ' R F + (R E + R L )(h FE + 1) 



(Ve 



Vcc)h } 



+ 



Thus, 



dl c = (hjE + l)(R E + Rl + Rf) 
dI CBO R F + (R E + R L )(h FE + 1) 



(8-26) 



a-c O 

input signal 




^.Vt 



FIG. 8-13. A biasing circuit with current and voltage feedback. 
If R F » (h FE + \)(R E + R L ), 



If R F « (h FE + l)(R F + /2 L ), 



1 + 



h FE + 1 
Rf 



Re + ^l 

When R E = in Fig. 8-13, only voltage feedback is being applied. It will 
be seen that R F provides the negative voltage feedback, and it also sets the 
base current bias. Consider the case where the Q point is established so that 
Vce = Vccl2. Neglecting I CBO and V BE , the following equations apply. 

j = zXce = zXcc 
R F 2R F 

I B (h FE + 1) = I E 

-V cc 



and 



/* = 



2R, 



182 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 



Hence, 



§= = £ = (A« + 1) 



(8-27) 



In a circuit containing a temperature sensitive device, it is usual to position the 
Q point at room temperature at about the midpoint of the load line as in this 
example since the maximum variation of the Q point in either direction is 
permitted. The stability factor is restricted to the value given by Eq. (8-26) 
with R E = and R F /R L = h FE + 1. Thus, 



h FK + 2 



l FE 

2 



(8-28) 



This value is high and so this simple circuit is rarely used. 

Figure 8-14 shows one way in which the stability factor can be reduced 
while keeping the base current at its previous value. An additional supply, 



input signal 




output signal 



FIG. 8-14. A bias stabilization circuit using voltage feedback. 

V BB and a resistance R ± are required as shown in the diagram. Notice also 
that R F has been split and that the junction point is grounded for alternating 
currents. This prevents negative feedback from being effective at the signal 
frequency and so keeps the a-c gain of the stage independent of the applied 
feedback. 



8-5 Power Amplifiers 

The final stage of an amplifier is usually required to deliver power into a re- 
sistive load. The main purpose of the output stage is the conversion of 
power from the supply into useful a-c power in the load. The stage is de- 
signed to maximize power efficiency and to minimize transistor dissipation. 
Current and power gain are not major design factors since the overall gain 
requirements of the amplifier can be satisfied by the design of the preceding 
stages. For this reason, the design of an amplifier usually starts with a 
consideration of the output stage. 



SEC. 8-5 



POWER AMPLIFIERS 



83 



Figure 8-15 shows the circuit and the characteristic curves of a CE 
transistor amplifying stage. The load line corresponding to the series resist- 
ance R L has been drawn on the characteristics. We distinguish three methods 
of operation of this type of stage according to the location of the quiescent 
point. Amplifying stages discussed so far have been operating in class A. 
In this condition, the transistor is operating during the whole of the input and 
output cycle and, if the device is linear, this results in distortion-free ampli- 
fication. For power amplifiers, where maximum output is desired, the posi- 
tion of the Q point for class A operation will be approximately midway up 
the load line as shown in Fig. 8-1 5(a). 

Class B operation is said to occur when the base circuit is biased so that 
very little base and collector current flows in the quiescent state. The tran- 
sistor will then conduct only during one half of the input wave and will be 



s 



Load line 




Class A 



Class B 
point 



Rl 



Input 
signal 



^lV r 



(a) (b) 

FIG. 8-15. (a) Idealized CE characteristics with load line, (b) CE 
transistor circuit with series resistive load (d-c bias not shown). 

cut-off during the other half-cycle. Two transistors, operating out of phase 
with each other, are required to reproduce the original input signal. This is 
known as a "push-pull" circuit. 

In class C operation of a single transistor stage, the base is biased so 
heavily that the transistor will conduct for less than 50 per cent of the signal 
cycle. The output consists of a reproduction of only a small fraction of the 
cycle of the input signal, and this introduces gross distortion. This type of 
operation is not suitable for untuned amplifiers but is used with high effi- 
ciencies for tuned amplifiers and oscillators. 

If the Q point is positioned halfway up the load line as shown in the 
idealized case of a class A amplifier in Fig. 8-1 5(a), it will be located at 



Ct 



(8-29) 



184 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

where K max and / max are the maximum permitted voltage and current for the 
transistor in the CE connection (see Sec. 8-6). When the maximum input 
signal is applied, the operating point swings up and down the load line so that 
the maximum current is / max , and the minimum current is zero since I CEO = 
for the idealized case. Assuming a sinusoidal signal, the peak-to-peak current 
is 7 max and 

'™ = ^ (8-30) 

The peak-to-peak swing of the a-c output voltage under these conditions is 
^max, giving 



The a-c power delivered to the load is 



/ V 

J max ' max (R_'\7 > \ 



Assuming that no distortion is produced, the power supplied to the stage is 
unchanged by the presence of the signal and is given by 

Ps = Ic.Vcc = Im&X 2 m&X (8-33) 

The efficiency of conversion of the d-c power from the supply to a-c power in 
the load is 

r, = f- = 25% (8-34) 

This is the maximum efficiency that can be obtained from a class A amplifier 
with a series resistive load. 



TABLE 8-2. Distribution of power dissipation between 
transistor and load. 

Total power 
supplied for 
Circuit Condition Transistor Power in load a maximum Efficiency 

dissipation transistor dis- 
sipation of 
a-c* d-c 50 units 



Series 
resistive 
load 


Quiescent 
Maximum output 


50 

25 




25 


50 
50 


100 
100 


Transformer- 
coupled 
load 


Quiescent 
Maximum output 


50 

25 




25 






50 
50 



25% 
50% 



* Useful power output. 



SEC. 8-5 



POWER AMPLIFIERS 



185 



When the stage is quiescent, the voltage across the transistor is equal to 
the voltage across the load, and so half the power is dissipated in the tran- 
sistor and half in the load. The total power supplied to the circuit and the 
d-c power dissipated in the load are unchanged when maximum signal is 
applied since the mean value of a sine wave is zero. Thus the transistor 
dissipation drops by the amount of the a-c power supplied to the load as 
shown in Table 8-2. 

In the preceding analysis, the load has been chosen to give maximum 
power output. This requires a load of 

"' (8-35) 



Rr = 



L 



If the maximum output is required, and yet the transistor has to work into a 
different load resistance, transformer coupling can be used. This circuit is 
shown in Fig. 8-1 6(b) where an external load R' L is coupled by a transformer 



^a-c load line 


N/ !>^d-c load line 


\ 




q\ 


1 \ 


i \ 



k 



Reflected load 



n : 1 



Input 
signal 








>§ 

1 o 

i 


o 




t 




V c , = V cc 
(a) 



(b) 



FIG. 8-16. (a) Idealized characteristics showing d-c and a-c load 
lines, (b) CE transistor circuit with transformer coupled load. 



of turns ratio n to the power amplifier. The reflected load into the collector 
circuit is n 2 R' L which can be made equal to V max II max by the choice of a suitable 
value for n. 

For direct currents, the resistance in the collector circuit is formed by the 
small d-c resistance of the primary windings of the transformer. The d-c load 
line is therefore almost vertical as shown in Fig. 8-1 6(a) and, for maximum 
output, the Q point is located on this line to give a quiescent current of 
^max/2. When the a-c signal causes the collector current to fluctuate, the 
operating point moves up and down the a-c load line shown on the diagram. 
The slope of this new line is — l/R L , where R L is the reflected resistance of the 



186 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

load across the primary of the transformer. The a-c and d-c load lines must 
intersect at the Q point since this is the operating point for zero signal. Under 
a-c conditions, however, the apparent resistance in the collector circuit is that 
reflected across the transformer giving an a-c load line as shown. 

The operating point will move along the a-c load line on both sides of the 
Q point. Thus the instantaneous voltage across the transistor will exceed the 
supply voltage, V cc , during a portion of the cycle. The reason for this is 
that the change in current through the transformer produces a "back-emf" 
across the transformer primary winding which adds to the supply voltage 
during part of the cycle. 

From Fig. 8- 16(a) by considering similar triangles, the Q point is located 
at 

I Gi = ^ V Cl = V CC = ^ (8-36) 

neglecting the small resistance of the transformer windings. When the 
transistor is operating at maximum capacity under class A conditions, the 
voltage and current swings are from K max to zero and 7 max to zero respec- 
tively. Hence, 

F "--|^[ and / ras = ^| (8-37) 

and the useful a-c power in the load is 

P = / rm sK rms = ^1™? (8 _ 38) 

The total power supplied to the circuit is 

p s = v cc r Cl = Ys^ (8 _ 39 ) 

The power efficiency, rj, for this circuit is 

r, = £ = 50% (8-40) 

r S 

Thus the maximum efficiency for the conversion of d-c power from the 
supply to a-c power in the load for the transformer-coupled class A amplifier 
is 50 per cent. The efficiency is double that of the previous case because no 
d-c power is dissipated in the load. Table 8-2 shows the power dissipation in 
the load and the transistor for this circuit assuming maximum transistor 
dissipation of 50 units. 

Whether the load is in series with the collector or transformer coupled, 
Table 8-2 shows that the transistor dissipation drops to half its quiescent 
value when maximum signal is applied. This means that the transistor 
operating temperature is a function of the signal amplitude for both types of 
stage. Biasing circuits which stabilize the transistor stage against tempera- 
ture variations, as discussed previously, are therefore a necessity. 



SEC. 8-6 POWER AMPLIFIERS: PRACTICAL LIMITATIONS 

8-6 Power Amplifiers: Practical Limitations 



87 



In an actual transistor, there are several limitations which must be taken into 
account. These are summarized here under the categories : maximum current, 
maximum voltage, maximum dissipation, minimum voltage (saturation). 

The output characteristics of a typical transistor in the CE connection are 
shown in Fig. 8-17. For high values of collector current, the curves for 



Maximum current limitation 



Low value of h t 




Maximum dissipation 



Maximum voltage limitation 

V CE 



FIG. 8-17. Output characteristics showing limiting values. 



constant base current increments are crowded closer together because of the 
drop in current gain as the collector current rises. Excessive distortion will be 
produced at peak currents if the collector current is raised indefinitely. 

The maximum voltage limitation is determined by the onset of collector 
multiplication, excessive surface leakage, or punch-through. Collector multi- 
plication is responsible for a marked rise in collector current as V CE is in- 
creased. It is caused by an ionization avalanche phenomenon initiated by 
collisions of fast moving charge carriers in the collector-to-base depletion 
region. When a high collector-to-base voltage is present, a carrier gains 
enough energy in this region to generate additional electron-hole pairs which 
add to the collector current. Consideration of Fig. 8-17 shows that consider- 
able distortion would be obtained if the transistor were operated in this 
region. Surface leakage is another undesirable effect in transistors and is also 
responsible for an unwanted increase in I c as V CE rises. Punch-through 
occurs when the depletion region at the collector end of the base is so large 



188 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 8 



that it meets up with the emitter-to-base depletion region and causes the 
active part of the base to disappear. The voltage at which this takes place is 
known as the punch-through voltage (see Sec. 9-11). 

The transistor should not be operated with a power dissipation greater 
than a certain value specified by the manufacturers. For a given ambient 
temperature, the maximum dissipated power may be stated as T Cl V c = 
constant. This equation has been plotted on Fig. 8-17, and the load line 
should always lie below this curve. A further discussion of this limitation is 
given in Sec. 8-7, Thermal Runaway. 




Maximum dissipation 



Maximum output voltage swing 
for 10% distortion 



FIG. 8-18. Positioning of the load line showing Q point and 
maximum output voltage swing. 



The part of the characteristics marked saturation forms another limitation 
for operation. Here, the collector-to-emitter voltage is insufficiently reverse 
biased for normal transistor operation. The voltage at which the transistor 
goes into saturation under given conditions determines the minimum allow- 
able collector-to-emitter voltage for negligible distortion (see Sec. 10-7). 

All these limitations must be taken into account by the circuit designer in 
deciding on a power amplifier circuit. For maximum power output and 
efficiency without too much distortion, the load and the collector supply 
voltage must be chosen to make good use of the output characteristics as 
shown in Fig. 8-18 for the class A single transistor circuit with a series 
resistive load. 

A transistor operating as a small-signal class A amplifier introduces a 



SEC. 8-6 



POWER AMPLIFIERS: PRACTICAL LIMITATIONS 



189 



small amount of distortion into the amplified signal because of nonlinearities 
in the transistor characteristics. This distortion is usually less then five per 
cent and can be tolerated in many applications. When large-signal operation 
takes place, the distortion is increased, and it may prove to be the limiting 




Time — * 




Flattening due 
to nonlinearity 

of input 
characteristics 



./ 



Undistorted value 



/ \ /-Flattening due to drop 



in (3 at high values of I c 




Undistorted \ 
value 



Flattening due to 
input distortion 



FIG. 8-19. Diagram showing effects of input and output dis- 
tortion in a CE stage. 

factor in the design of the output stage. It is found, in practice, that if a large- 
signal CE amplifier is driven from a low resistance source, distortion is 
minimized. This is the result of partial cancellation of the separate distortions 
occurring because of the nonlinearities in the input and output characteristic 
curves of the transistor. This is illustrated diagrammatically in Fig. 8-19. 



8-7 Thermal Runaway 

The junction temperature T t of a transistor is determined by the ambient 
temperature T A and the power dissipation P of the transistor according to the 
equation 

T i = T A + KP (8-41) 



190 



SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 



where K is known as the thermal resistance. From this equation it can be seen 
that 



„ AT, AP 1 

K =AP ° r Af, = K 



(8-42) 



and so K is the junction temperature rise for unit increase in the power dis- 
sipation and has the dimensions °C (watts) -1 . Alternatively, l//£can be ex- 
pressed as the increase in power dissipation necessary to raise the junction 
temperature by 1° C. K is a function of the thermal conductivity of the 
materials used in the transistor, the shape and size of the transistor, and the 
method of mounting that is used. 

If the collector current and voltage in a CE stage are fixed, the power dis- 
sipation is constant and is approximately I C V CE . The junction temperature T, 

will attain its equilibrium value given 
by Eq. (8-41) within a few minutes of 
switching on the circuit, assuming that 
the ambient temperature remains con- 
stant. The maximum allowable power 
dissipation for the transistor at a given 
ambient temperature (usually 25° C) will 
usually be given by the manufacturers in 
the data sheet. If the ambient temper- 
ature is increased, the maximum dis- 
sipation must be reduced according to 
Eq. (8-41) and the curve of Fig. 8-20 
applies. This curve shows the value of K. 
Most CE amplifiers employ a 
biasing circuit which compensates for 
changes in quiescent operating point 
with temperature variation. These 
biasing circuits operate by negative feedback and inevitably introduce d-c 
power losses in the stage. In output stages where power efficiency is of im- 
portance, biasing circuits of only marginal stability are frequently used to 
reduce these losses, and considerable changes in I c may occur as the tem- 
perature is increased. Under these conditions, the power dissipation, P, is 
no longer a constant but is a function of junction temperature, T,. This 
function depends upon the properties of the transistor and also upon the 
external circuit. 

It is possible to connect the transistor in a circuit so that the junction 
temperature rises continuously and eventually exceeds the safe limit. The 
effect is that a slight rise in temperature produces an increase in the power 
dissipation of the transistor, and this, in turn, increases the temperature of 
the junction. When the effect is cumulative, it is known as thermal runaway 
and is obviously undesirable. 




25° 50° 100* 

Ambient temperature in °C 

FIG. 8-20. Maximum power dissipation 
of a transistor as a function of ambient 
temperature. 



SEC. 8-7 THERMAL RUNAWAY 191 

From Eq. (8-42), the increase in the rate of transfer of energy from the 
junction (AP) for unit temperature rise (AT } = 1° C) is l/K. This energy is 
transferred through the body, the case, and the mountings of the transistor 
and is a constant for a given type of mounting. However, the actual increase 
in energy dissipated in the transistor because of unit temperature rise is 
determined by the type of circuit and the characteristics of the transistor. If 

AT, > K (8 " 43) 

more energy is dissipated in the transistor than can be transferred away from 
the junction, and so this is the condition for thermal runaway to occur. 
For the typical circuit of Fig. 8-9, 

P = f C VcE 

= Ic(Vcc - IcRl) (8-44) 

The condition to prevent thermal runaway, from Eq. (8-43) is 

AP 1 

25j < I (8 " 45) 

or, by differentiating Eq. (8-44), 

^(V CC -2I C R L )^± (8-46) 

Equation (8-46) shows that for a given circuit, better stabilization of 7 C , a 
larger load resistance, or a reduction in K might be used to prevent thermal 
runaway. Al c /ATj may be expressed as S(dI CBO l8Tj), where SI CBO ldTj is given 
in Fig. 8-6 and Eq. (8-9). Thus, reducing the stability factor, S, may be 
sufficient to prevent thermal runaway. The collector load R L also plays a 
critical part in conditions for thermal runaway. Power output stages which 
are transformer coupled to the load are particularly susceptible to runaway 
because of the low d-c resistance of the transformer windings. Adding re- 
sistance in the collector lead, if it can be tolerated, may be sufficient to 
prevent this occurring. Finally, the value of K may be decreased by mounting 
the transistor with good thermal contact to a large, well-cooled metallic body 
known as a "heat-sink." The value of K when an infinite "heat-sink" is used 
is often given in the data sheet of the transistor. 

8-8 Multistage Amplifier 

Figure 8-21 shows the circuit of a CE two stage, small-signal amplifier where 
the two stages are identical. Transistors T ± and T 2 are assumed to have the h 
parameters given in Table 7-4. Since the values of the parameters change 
with quiescent operating point, the emitter current and the collector voltage 



192 



SINGLE AND MULTISTAGE A-C AMPLIFIERS 



CHAP. 8 



will be assumed to have the values as specified for the transistor in the table. 
Typical variations in h parameters as the emitter current and collector voltage 
are varied are given in Figs. 7-10 and 7-11. 

The performance quantities of the single stage have been given previously 




ov rr = -12v 



O Output 



Input 



FIG. 8-21. A two-stage, small-signal amplifier. 

in Table 7-2 in terms of a source resistance, R s , and a collector load, R L . 
From these tabulated values, the current gain of transistor T 2 is given by 



l fe 



49 



1 + h oe R 7 1 + (5 x 10- 5 )10 4 



« 32.5 



(8-47) 



In the multistage amplifier, however, the interstage coupling network must 
also be considered as shown in Fig. 8-22. I x and I 2 are the alternating currents 
supplied by the collector of 7\ and supplied to the base of T 2 respectively. The 
impedance of the V cc supply to alternating currents is very low so R 5 and R 6 
are effectively in parallel. Assuming that the reactance of C 2 can be ignored 



OVr 



h 



(a) 



(b) 



FIG. 8-22. (a) Interstage coupling network, (b) Equivalent circuit 
at high frequencies. 

at the frequency of operation of the amplifier, the a-c equivalent circuit re- 
duces to resistances R 3 , R 5 , and R 6 in parallel, as shown in Fig. 8-22(b). 
Writing R for the value of this parallel combination of resistances, 



h R + R i2 

For this circuit, R = 4760 ohms and R i2 has still to be determined. 



(8-48) 



SEC. 8-8 MULTISTAGE AMPLIFIER 193 

The load resistance, R 7 , of the second stage is 10,000 ohms and so the 
input resistance of this stage is 

A he R 4- h 

R i* = 1 f\ P = 1480 ° hmS ( 8 ~ 49 ) 

from Table 7-4. Thus, from Eq. (8-48), 

^ a 0.77 (8-50) 

Transistor 7\ is operating into a load formed by the parallel combination of 
the four resistances i? 3 , i? 5 , i? 6 , and R h as shown in Fig. 8-22. For the given 
values, the effective load is 1130 ohms. The current gain for transistor T 1 is, 
therefore, 



hfe 49_ 

' 1 + h oe R Ll 1 + (5 x 10" 5 )1130 



K - , ,T D = 1 , ,c„ ,n-6M,™ « 47 (8-51) 



In the input circuit, the current flowing into the base lead of the first 
transistor is only a fraction of the input current through R s . The input 
resistance of 7\ is 

Ahe D > L 

R h = i y P = 192 ° 0hmS ( 8 ~ 52 ) 

1 + /7 oe A Ll 

By analogy with Eq. (8-48), we can write 

A. tf' 



where R' is the parallel combination of R u R 2 , and /? fl . Now,/?' = 9100 ohms, 
and so 

r bl 9ioo 



/, 9100 + 1920 * °' 82 (8 " 54) 

The total current gain through the circuit from the input lead to resistance 
R 7 is 

J f = 32.5(0.77)(47)0.82 = 960 (8-55) 

The stability factor for each transistor stage can be obtained from 
Eq. (8-23) 

1 + R B /R E 



1+ * 



(8-23) 



(h FE + 1)^ 

/**.£ will be specified by the manufacturers, and we will assume that it has the 
value 40. R B is formed by R ± and R 2 in parallel giving R B = 9100 ohms. 
Hence, 

9100 

+ 41(1000) 



194 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

8-9 Frequency Response of the Multistage Amplifier 

The frequency response of the typical multistage amplifier, shown in Fig. 8-21, 
is determined partly by the circuit elements and partly by the inherent fre- 
quency limitation of the transistors. At very low frequencies, the reactances 
of capacitances, C\ and C 2 can no longer be ignored in the interstage coupling 
network, and they reduce the magnitude and affect the phase of the current 
which is flowing. The reactances of capacitances C 3 and C 4 also increase as 
the frequency is reduced and, as a result, the emitter resistances /? 4 and R 8 
are no longer adequately bypassed. This causes an additional decrease in 
gain. 

At high frequencies, the current gain of the transistor decreases because 
of the time taken for the carriers to pass across the base region. The fre- 
quency at which the short-circuit CE current gain has dropped to 0.707 of 
its low frequency value is known as the beta-cutoff frequency, f s . In the CB 
connection, the short-circuit current gain drops to 0.707 of its low frequency 
value at the alpha-cutoff frequency, f a . The relationship between the two 
quantities is, 

J a ~ "■fejfi 

and since h fe is usually between 40 and 150, f B is much less than/ a . A typical 
inexpensive alloy-junction transistor has an alpha-cutoff frequency of 
5 Mc/sec and h fe = 50. For this transistor, /# is only 100 kc/sec and this con- 
stitutes a fundamental limitation on the use of this amplifier at high 
frequencies. (See Chap. 10 for a discussion of this point.) 

In addition to this effect, the time constant formed by the collector load 
resistance and the output capacitance of the transistor produces a loss in 
gain at high frequencies. Because of this, the gain of the stage will have 
dropped to 0.707 of its low frequency value at a frequency given by 

/= ' 



C oft /v/ 



where C oe is the collector output capacitance in the CE connection. More 
commonly, the output capacitance in the CB connection (C ob ) is specified. 
These two quantities are related according to 

C oe ~ rlfe^ob 

In many circuits, the beta-cutoff frequency is lower than the frequency at 
which the output time constant causes a loss in gain. When a high value of 
collector load, R L , is used, however, the time constant may have to be 
considered. 

The high frequency response of the CE stage is discussed in Sees. 10-2 
through 10-5 where the hybrid-77- model of the transistor is analyzed. 



BIBLIOGRAPHY 195 

BIBLIOGRAPHY 

Cote, Alfred J., Jr. and J. Barry Oakes, Linear Vacuum-tube and Transistor 
Circuits, New York: McGraw-Hill Book Company, Inc., 1961 

Fitchen, Franklin C, Transistor Circuit Analysis and Design, Princeton, 
N.J.: D. Van Nostrand Company, Inc., 1960 

Greiner, R. A., Semiconductor Devices and Applications, New York: McGraw- 
Hill Company, Inc., 1961 

Hurley, Richard B., Junction Transistor Electronics, New York: John Wiley 
&Sons, Inc., 1958 

Joyce, Maurice V. and Kenneth K. Clarke, Transistor Circuit Analysis, 
Reading, Mass.: Addison- Wesley Publishing Company, Inc., 1961 

Pullen, Keats A., Handbook of Transistor Circuit Design, Englewood Cliffs, 
N.J.: Prentice-Hall, Inc., 1961 

PROBLEMS 

8-1 A transistor having the output characteristic shown in Fig. 8-5 is 
operated in the CE connection with a resistive load in the collector lead 
of 333 ohms and with V cc = — 10 volts. If I B = 200 /xa, what is the 
quiescent operating point? If an alternating current of 200 /xa peak to 
peak is applied to the base, what is the value of the alternating output 
current? What is the maximum input alternating current that could be 
applied without excessive distortion? 

8-2 Determine the value of h fe for the quiescent operating point of the 
transistor in Prob. 8-1. If the input alternating current is 200 /xa peak 
to peak, find the mean value of h oe from a knowledge of A t and h fe . 

8-3 Explain why the output characteristics of germanium and silicon 
transistors are about the same although there are significant differences 
in their respective input characteristics. 

8-4 A pnp transistor is operated in the CE connection with a load resistance 
R L in the collector lead. A high resistance R ± is connected between the 
negative terminal of the collector supply battery and the base terminal. 
If Vcc = — 20 volts and a = 0.98, calculate the values of R L and R 1 
to make V CE = 10 volts and I c — 2 ma, making any reasonable assump- 
tions considered necessary. What is the major limitation of this circuit? 

8-5 For the circuit of Fig. 8-11 find (a) the stability factor, S, (b) S h , (c) 
dlc/d^BE given that R x = 20,000 ohms, R 2 = 60,000 ohms, R E = 2000 
ohms, R L = 3000 ohms. Assume that a = 0.98, I CBO - 3 /xa, V BE 
= —0.2 volts, and V cc = — 16 volts. 

ans. S = 6.5, S h = 4.3 x 10~ 6 amps, 

dI c ldV BE = -4.25 x 10" 4 amps volts" 1 



196 SINGLE AND MULTISTAGE A-C AMPLIFIERS CHAP. 8 

8-6 Using characteristic curves and equations where necessary, show how 
the location of the quiescent operating point is temperature sensitive. 

8-7 A transistor is connected as in Fig. 8-11. If V cc = — 20 volts, R L 
= 2000 ohms, h FE = 50, and S is to be 12, calculate the values of the 
resistances to give as large an output voltage across R L as possible 
without distortion. What is the minimum value for C E if the circuit is 
to be used as an audio amplifier? 

8-8 The transistor of Fig. 8-20 is operated in a CE connection where the 
stability factor S is 20 and there is no load resistance in the collector 
lead. I C Bo = 10 j^a at 25° C and V cc = -25 volts. What is the critical 
value of collector current above which thermal runaway will occur? 

ans. 13.2 ma 

8-9 Two transistors are connected in cascade to supply a 1000 ohm load. 
Each has the following parameters: h ib = 30 ohms, h fe = 49, h rh 
= 10~ 4 , h ob = 10 -6 mhos, (a) Find the current and power gain of the 
circuit if both transistors are connected in the common base con- 
figuration, (b) Find the current and power gain if the first transistor 
is connected in a CB configuration and the second is in a CE connection. 
Design a suitable interstage coupling network in both cases. 



PART 



3 



Further 

Theory of 

the Transistor, 

and the 

Tunnel Diode 



9 



Physical 

Characteristics 

of the Transistor 



The continuity equation, which was derived in 
Chap. 4, is applied here to the base region of the 
transistor. First, a solution is found for the charge 
and current density as a function of distance in the 
base region when the transistor is operating under 
d-c conditions. This solution leads to a derivation 
of the transport factor and a discussion of the 
components of alpha. Then, the transistor is con- 
sidered to have an applied alternating emitter 
current, and it is shown that the solution of the 
continuity equation under a-c conditions results in 



200 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



an analogy with an RC transmission line. The transmission line is replaced by 
a lumped RC network, and this circuit leads to the useful hybrid-™ representa- 
tion of the transistor. 

The effect of electric field in the base region is considered for high emitter 
currents in the diffusion transistor (Sec. 9-7) and for the important case of 
the drift transistor (Sec. 9-8). The chapter concludes with a brief review of 
other transistor characteristics. 



Solution of the Continuity Equation 
in the Base Region for Direct Currents 



The hole density in the base region of a pnp transistor is given by the con- 
tinuity equation (4-9), 

dp = PnO 
dt 



*-ȣfr/> + D,g 



(9-1) 



When only direct currents are flowing, dp/dt is zero. For a transistor having 
abrupt junctions, where the resistivity of the base region is constant, the 



y'Edge of emitter-base 
* depletion region 



rEdge of collector-base 
depletion region 




P r ^0 



FIG. 9-1. Hole density in the base region of a. pnp transistor. 



passage of charge across the base under small-current conditions is the result 
of diffusion only, and p v d(p£)\dx is zero. The continuity equation then 
becomes 

d 2 p p - p n0 



A 



dx* 



or, since D p r p = L 2 , where L p is the diffusion length, 

d 2 p p - PnQ 
dx 2 Ll 



(9-2) 



(9-3) 



SEC. 9-1 SOLUTION OF THE CONTINUITY EQUATION 201 

A solution of this equation is of the form, 

p - Pn0 = A exp ( - ^J + B exp (^) (9-4) 

where A and B are constants. These constants can be determined from the 
known values of p at the emitter-base and collector-base junctions. From 
Fig. 9-1, at the emitter-base junction, 

x = 0, p = Pe = Pn0 exp (~p) (9-5) 

where p e » p n0 with V EB » kT/e, and at the collector-base junction, 

x = W, p = p c « (9-6) 

where V CB « -kT/e. 

At the collector end of the base region we note that both p c and p n0 are 
very small quantities. From Eqs. (9-4) and (9-6), therefore, 

A eXP \L~) + B CXP \T) = Pc ~ PnO ~0 

A= -Bexp(Y) (9-7) 



giving 



Thus, 



p - p n0 = - B [exp (^) exp (-^) - exp (g)] (9-8) 

At the emitter end of the base region, substituting Eq. (9-5) into Eq. (9-8), 
Pe = -£[exp(^)- 1] 



Hence, 



B = — (9-9) 

l-exp(2^/L p ) Ky y) 



Substituting this value for B in Eq. (9-8) 

-* ah (?H(if) --»(£)] <-»> 



P Pn0 l-exp(2^/L WL 



The hole current density at any point in the base region is 



-eD & 
6JJp dx 



Thus, 



/+ = -eA>A 



L p [l -exp(2^/L, 



202 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

This analysis is valid for direct currents and also for very low frequencies. 
We now define / c + to be the hole current density leaving the base region at 
the base-collector junction where x = W. Similarly, J r e + is defined as the hole 
current density entering the base region at the emitter-base junction where 
x = 0. The ratio of these currents is known as the transport factor, p*, 
where the subscript zero indicates that this is a low frequency value. From 
Eq.(9-ll), 

J c + exp(W/L p ) + exp(W/L p ) 



R = 



J: exp(2^/L p )+l 

= cosh (W/L p ) (9-12) 



exp(W/L p ) + Qxp(-W/L p ) 

It is desirable to make £j as near to unity as possible since it is a principal 
component of alpha (see Sec. 9-2). This is achieved by making the base 
region so thin that W « L p . In this case, exp (W/L p ), can be replaced by the 
first three terms of the series 

Lip £l-j p 

and exp (— W/L p ) can be replaced by the first three terms of the series 

_ W W^_ 

Li p £Li p 

Rewriting Eq. (9-12) in this approximate form, 

2 1 W 2 

P* ~ 2 + W 2 /L 2 = 1 + W 2 /2L 2 P ~ l 'IL 2 . (9 ~ 13) 

9-2 Components of Alpha 

The transport factor, j8j, which was derived in the previous section, is one 
of the three factors which determine the short-circuit current gain of the 
transistor at low frequencies, a . We can write, 

«o = /5Jyo8 (9-14) 

where y is the emitter efficiency at low frequencies, and S is the collector 
multiplication factor. From Fig. 9-2, the total current density entering the 
emitter is J e , the hole current into the base region is designated 7 e + , and the 
electron current is /". The emitter efficiency as defined in Sec. 5-3B is 
7o = Je IJe- Only the hole current, //, takes part in transistor action and this 
current is reduced by recombination in its passage across the base. The 
transport factor takes this recombination loss into account and so the hole 
current reaching the collector-to-base depletion region is B*J? = p*y J e - 

The collector multiplication factor, 3, is the ratio of the current leaving the 
collector region to the hole current entering from the base. This factor can 



SEC. 9-2 COMPONENTS OF ALPHA 203 

exceed unity, even for low voltage operation, in transistors where the resis- 
tivity of the collector region is high (for example, grown-junction types). The 
passage of a high current through the region results in an electric field of 
signal frequency, $ = J? p, where p is the resistivity of the material. This 
field modulates the thermally generated minority carrier current and produces 
a flow of electrons in the opposite direction to the hole flow. Thus, the col- 
lector current, J c , is higher than the hole current from the base, J c + , and 8 is 
greater than unity. S will increase with temperature since the number of 
minority carriers rises with temperature. 

When the collector-to-base voltage is high, an avalanche mechanism can 
develop which enhances the value of S. Fast moving positive holes may be 



Hole current density 
into base J e + = yj 



Total current density 

entering emitter 

region J e 



Collector current density 
J c =(3$yhJ e = a J e 
Hole current density 
leaving active 
base region 



Electron current density 

into base (electrons 

moving in opposite 

direction) 

I 



Emitter Base Collector 

FIG. 9-2. Hole and electron currents flowing in the transistor as 
a result of an emitter current, J e . 

sufficiently energetic to produce secondary ionization in the collector deple- 
tion region and increase the total current. This increase in 8 is a limiting 
factor in power transistor operation (see Sec. 8-6 and Fig. 8-17). 

9-3 The Intrinsic Transistor 

One of the purposes of transistor analysis is to devise a simple two port net- 
work, which will represent the device under given conditions. The transistor 
model can be split into two parts as shown in Fig. 9-3, (a) the intrinsic tran- 
sistor and (b) the external elements. The intrinsic transistor, which is shown 
here as a "black-box," includes elements which represent the diffusion effects 
in the base region and other internal transistor characteristics. The external 
elements shown are the collector and emitter junction capacitances and a 
resistance, r bb >. The two junction capacitances can be simply represented as 
two single capacitances connected to a common point on the active base 
region, B'. Between B' and the external base terminal B is the resistance r bb >. 
This is known as the base spreading resistance and is the ohmic resistance of 



204 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



Eo- 



C,.5fc 



Intrinsic 
transistor 



-oC 



^c,- 



OB 
FIG. 9-3. Transistor model showing the intrinsic transistor. 

the semiconducting material between the active base region and the base 
connection (see Fig. 9-4). There are also ohmic resistances associated with 
the emitter and collector leads, but these can usually be neglected. The base 
spreading resistance is of importance since it limits the high frequency and 



Collector 



Base contact 
(ring-shaped) 




Inactive base 
region 



Inactive base 
region 



Emitter 



FIG. 9-4. Schematic diagram of an alloy-junction transistor 
showing the inactive part of the base region which is mainly 
responsible for the base spreading resistance. 

pulse operation of the transistor (see Chap. 10), and the manufacturer will try 
to reduce it to as low a value as possible. Values for r w range from about 
ten to several hundred ohms. 



9-4 The Continuity Equation 

in the Base Region for Small Alternating Signals 



Consider that a small alternating voltage of value V eb exp (yW) is impressed 
on the direct emitter-to-base voltage V EB . The instantaneous total voltage is 
then, 

v eb = Veb + K e5 exp(yW) (9-15) 

Assuming V EB » kT/e, the injected hole density in the base close to the base- 
to-emitter depletion region is given by 



SEC. 9-4 THE CONTINUITY EQUATION 205 

Pe = Pno exp Wj, \v EB + V eb exp (jtot)\ j 

= Pno [exp (^JJjexp [t^ V eb exp ( 7 W)]} (9-16) 

Putting in the condition that V eb « /ciy^, i.e., V eb is restricted to well below 
25.8 millivolts at 25° C, we can write exp [(e/kT)V eb exp (yW)] in the approxi- 
mate form 1 + (ejkT)V eb exp (yW). Thus, 

/7 e = p n0 exp (^p j [l + -£fV eb exp (yW)l 

= p n0 exp (^) + /7 n0 [exp (^)] [t^ ^ exp (yW)] 

= /? ei + j p e2 exp(yW) (9-17) 

where /? ei = direct component of hole density at x = caused by voltage 

p e2 = maximum value of the alternating component of the hole 
density at x = as a result of the alternating voltage V eb . 

When the transistor is operating with the collector-to-base junction 
reverse biased, the hole density at the collector end of the base region is 
approximately zero. Both the direct and alternating components of the hole 
density are reduced to zero at the collector junction and, in general, we can 
write the hole density at any point in the base region in the form 

p = Pi + p 2 exp (Jut) (9-18) 

where the subscript 1 indicates a direct quantity and the subscript 2 denotes 
an alternating quantity. Substituting Eq. (9-18) into the continuity equation 
(9-1), and neglecting the drift term, 

/• x PnO — Pi — P2 eXD(/W) _ d 2 D 1 _ d 2 _ /• m 

ja>p 2 exp (jwt) = ^ — /* FU } + D p -£± + D p -^ [p 2 exp (jcot)] 

(9-19) 

This equation can be split into two parts: a d-c equation which is similar to 
Eq. (9-2), 



= PnO ~ Pi , r, d<1 Pl 

and an a-c equation, 



= £2S £± + D v -^ (9-20) 

t„ p dx 2 



jwp 2 exp (yW) = -- exp (Jcot) + D p — 2 [p 2 exp (Jut)] 



or, by dividing through by exp (jcot) 

r n + ^ dx 2 



jo ,p 2 =-Pl + Dp &£l (9-21) 



206 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 



Rdx 




Cdx 



FIG. 9-5. Leaky RC transmission line. 

Equation 9-21 is similar to the equation which can be derived for the RC 
transmission line shown in Fig. 9-5. The series resistive element has a re- 
sistance R per unit length of the line. The parallel resistance is most con- 
veniently represented by its conductance G per unit length, and the capacitance 
has a value C per unit length. For an element of length dx, and for alternating 
currents and voltages defined in Fig. 9-5, 



/ = 



]_dV 
R dx 



and, by differentiating with respect to x, 

dl = 1 d 2 V 

dx R dx 2 



(9-22) 
(9-23) 

(9-24) 

(9-25) 

(9-26) 
(9-27) 



Equation (9-27) is an equation for the maximum value of the alternating 
charge on the line as a function of distance. Equation (9-28) is an equation 
formed by multiplying all the three terms of the a-c continuity equation, 
Eq. (9-21), by the electronic charge, e. 



Now, 




dl = 


= (G + ja 


QVdx 


so 




dl _ 
dx 


= (G + ja 


>C)V 


Combining 


Eqs. (9-23) 


and (9-24), 








JcoCV 


1 d 2 V 

= -GV + — — — 

KJV + R dx 2 


The alternating charge 


on the capacitance 


is given by 






Q = 


CV 




Thus, 




j"Q = 


-§ e+ 


1 d 2 Q 
CR dx 2 



jojep 2 = 



ep 2 



+ A 



Z\ep 2 ) 

dx 2 



(9-28) 






SEC. 9-4 



THE CONTINUITY EQUATION 



207 



Comparing Eqs. (9-27) and (9-28), it is seen that an analogy exists between 
the alternating charge in the base region of the transistor (ep 2 ) and the 
alternating charge on the line (Q). 

The graphical solution of the two equations is shown in Fig. 9-6. The 
solution will not be derived here, but the form of it can be seen by considering 
the limiting values of the hole density at the two ends of the base region. At 



Charge in active 

base region 

ep 



(a) 




Distance x 



Maximum alternating 
charge on capacitance 

Q 

(b) 



Q = 



\ 



Distance x 



W 



FIG. 9-6. (a) Charge in the active base region of the transistor 
showing maximum value of the alternating charge as a function of 
distance, (b) Maximum value of alternating charge on analogous 
transmission line as a function of distance. 



the emitter-base junction, x = and/? ei and/? e2 are the direct and alternating 
components of hole density having the values given in Eq. (9-17). At the 
collector-base junction, x = W and the direct and alternating values of 
the hole density, p Cl and/? C2 , are both approximately zero. This assumes that 
the collector-to-base junction is reverse biased with V CB « —kT/e. The 
analogous condition to this on the transmission line representation is that 
Q = at x = W. The line is, therefore, short-circuited at its far end. 

Between x = and x = W on the transmission line, the conductance, G, 
allows a fraction of the charge to leak away. Thus, the current flow through 



208 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



G represents the loss of holes in the base by recombination. If recombination 
in the base is neglected, G must be made zero. 

9-5 Base Width Modulation: Transmission Line Analogy 

The previous section has shown that there is an analogy between the alternat- 
ing charge at any point in the base region of a transistor and the alternating 




R 

VW 



Ic = O! I e 




Short circuit 
atx=W 



FIG. 9-7. Alternating currents and voltages on the line. 

charge on the transmission line of Fig. 9-5. It has been shown that there is a 
short circuit at the position x = W corresponding to the condition that the 
alternating hole density at the collector-to-base junction is zero. The analogy 
is carried further in Fig. 9-7 where the alternating currents and voltages 
which are known to be present in the transistor are applied to the line. The 
current entering the transmission line is the alternating emitter current, 7 e , 
and the current passing through the short circuit at low frequencies is the 



Alternating- 
current 
generator 



\J 



'ee t 



Alternating voltage 
drop, V C = I C R L 



-±-V, 



FIG. 9-8. Alternating voltage drop across load resistance, R L . 

collector current I c . The amount of recombination in the base region is 
accounted for by the presence of the conductance, G, and so the collector 
current, I c , can be written as j8j/ e . When the emitter efficiency and the 
collector multiplication factor are taken to be unity, then jSj = a and 

So far, it has been assumed that the collector-to-base voltage is a constant. 
This is not a valid assumption, in practice, since the presence of a load in the 



SEC. 



BASE WIDTH MODULATION 



209 



collector circuit introduces a voltage drop between the collector supply and 
the collector terminal as shown for the CB circuit in Fig. 9-8. An alternating 
voltage between the collector and base terminals is produced here by the 
alternating collector current flowing through R L . In general, therefore, we 
have to consider the effect of a collector modulating voltage on the trans- 
mission line analogy. This can be done by recalling that the collector-to-base 
voltage determines the depletion width at the collector end of the base region, 
and hence the active width, W, of the base region itself. In the analysis which 
follows, we shall distinguish between the active base, B', and the base terminal, 
B. The alternating voltage that is responsible for the change in base width is 
that between the collector and the active base and is designated V cb >. The 
voltage, V b > b , is across the base spreading resistance, r w , of Fig. 9-3 and the 



Base 



d-c plus maximum 

a-c hole density 

P1+P2 



dW 




Collector - 

to-base 

depletion region 



W W+dW 

Distance — ► 

FIG. 9-9. Enlarged view of collector-to-base junction. 



sum of these two voltages is V cb which is across the external collector and base 
terminals. In some cases, V Vh can be neglected and V cb > and V cb assumed to 
be identical. The correct notation will be retained in this section since the 
intrinsic transistor is being discussed. 

In Fig. 9-9 a much enlarged view of the edge of the base-to-collector de- 
pletion region is shown. When no alternating emitter current is applied, the 
active region of the base ends at x = W as shown. When the alternating 
current reaches its maximum value, the upper curve of hole density applies 
and the active region of the base extends out to W + dW. Making the 
assumptions, dW « W, p e2 « p ei and neglecting recombination, the slopes 
of the two hole density curves given in Fig. 9-9 are equal and are given by 

dp 



— _t.fi for the d-c case 
dx W 

The boundary conditions, 

Pox + Pc 2 = at x = W + dW 



(9-29) 



(9-30) 



210 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

can be transformed to 

p 2 =Pac at* = W (9-31) 

by making use of the relationship 



Pac 

dW 



dp = Pe± 

dx W 



Thus, 



Pac^dW 



(9-32) 



The boundary condition of Eq. (9-30) means that the effective base width 
is a function of the collector voltage. The transformed boundary condition 
given in Eq. (9-31) shows that the active base width may be considered to be 




Ic 



VW 



^Charge - 

Qac 



Fixed length 
W 



FIG. 9-10. Voltage generator, /xK cb ', representing the effects of 
base-width modulation on the transmission line. 



fixed at the value Wand then the hole density at x = Wis given by Eq. (9-32). 
Considering a junction of unit area, we can write the direct current entering 
from the emitter as 



Hence, 



or 



h 


= -eD * 




= eD p 


Pe x 

w 


Pac 


eD p 


dW 




h 


dW d] 




eD 


P dV CB ' 


Wdc 




dW v„. 



(9-33) 



(9-34) 
(9-35) 



CB' 



Returning to the transmission line analog, we see that the boundary con- 
dition of Eq. (9-30) leads to a transmission line with a length that is a 



SEC. 9-5 BASE WIDTH MODULATION 21 I 

function of the collector-to-base voltage. This is an inconvenience which can 
be overcome by using the transformed boundary condition of Eq. (9-31) 
which fixes the line length at Wand introduces a charge Q ac = ep ac a.tx = W. 
This is equivalent to replacing the short circuit by a voltage generator of 
value fiV C b', as shown in Fig. 9-10, where 

vVcvC = Q ac = ep ac (9-36) 

epac h dW 

^cvr, = CD p dv7, (9 " 37) 

from Eq. (9-35). To find the value of the capacitance, C, we write 



c = § < 9 - 38 > 



and at the emitter, 



epe 2 

>2 



jfrP«y„ exp (^') (9-39) 



from Eq. (9-17). Hence, 



c _dQ__^ leV EB .\ 

C ~ dV eb . ~ kT Pn0 CXP \ kT J 



from Eq. (9-33). Thus, 



M = 



kT Pe, kTDp V w; 

I E dW kTD p 
D p dV CB . eWI E 

kT dW 

eWdVrn' 



(9-41) 



For a typical germanium alloy-junction transistor, the fractional change in 
base width is of the order of two per cent for every volt change between 
collector and base. At 300° K, 

kT dW 
\i = ^jy~ ~ °- 026 (°- 02 ) - 5 x 10" 4 (9-42) 



9-6 The Hybrid-n Representation 

The circuit of Fig. 9-1 1 shows the transmission line representation of the in- 
trinsic transistor and includes the current generator, I c , feeding the collector 
and active base terminals. It is not yet in a practical form because no account 
has been made of the power which must be drawn from the collector and 



212 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 





FIG. 9-1 I. Transmission line analogy showing collector circuit. 

active base terminals, C and B', to supply the voltage generator, /xK cb .. 
However, this circuit can be manipulated into one of several forms which are 
suitable for circuit analysis. In this section, we will derive the hybrid-^ 
circuit which is used in Chap. 10. (See Sec. 10-1 for a discussion of this 
circuit.) 

The transition between the transmission line representation and the 
hybrid-77 circuit is given in Figs. 9-11 through 9-16. The equivalence of the 
two circuits can be demonstrated by standard methods of circuit' analysis but 
the process is long and involved. In this account, step-by-step changes will 
be made in the circuit and approximations will be introduced where desirable. 

In the first place, we note that the collector current, I c , is only slightly 
smaller than the emitter current, I e , since a ~ 1. Therefore, there is only a 
small recombination loss of current in the base region. This means that the 
conductance, G, of Fig. 9-11 is low and its resistance is high. Because of this, 
it is possible to draw a simple circuit which approximates to the transmission 



Eq 



R 2 

A/W 



Ic 



rS zk Cl rS ic 2 0/xVc* 



K, 



iBT 



(a) 



R 2 £ 

AAA 



Rx< ^ 



0mK 6 . 



/„" 



^s/m> ^r»c 2 v cb . 



6B' 



(b) 



FIG. 9-12. (a) Replacement of transmission line by -n equivalent 
circuit, (b) R 3 and C 2 transferred to output circuit. 



SEC. 9-6 THE HYBRID-* REPRESENTATION 213 

line. In Fig. 9-1 2(a), a 77 circuit is used in place of the line. i?i and R 3 now 
account for the recombination loss of current, and R 2 is the equivalent series 
resistance of the line. C x and C 2 are the capacitance elements across the 
parallel resistances as before. R ± — R 3 and C 1 = C 2 and so the recombina- 
/tionjoss and the effective capacitance of the line have been divided into two 
equal parts; one at each end of the line. 

In Fig. 9-1 2(a), the power to operate the voltage generator, pV cb >, is 
obtained from the collector-to-base voltage, V cb >. Since R 3 and C 2 are con- 
nected in parallel across the constant voltage generator, they have no effect on 
the input circuit. Thus R 3 and C 2 can be transferred to the output circuit 
where they become of value R 3 /fx and i*C 2 , respectively, in order to keep the 
power and phase requirements the same. The circuit now appears as in 
Fig. 9- 12(b). 

Before proceeding further, the values of the resistances and the capacit- 
ance will be found. When the output is short-circuited to alternating currents 
(Kb' = 0), the collector current, l c , the emitter current, I e , and the base 
current, I b , are given by the equations, 

h = <*h (9-43) 

and I b = I e {\ - a) (9-44) 

Furthermore, at low frequencies where the effects of capacitance can be 
neglected, the low frequency value of alpha, a , can be used and 

?f = r e (9-45) 

where r e is the emitter resistance which is the incremental resistance of the 
forward-biased emitter-to-base junction defined in Sec. 4-7. In Fig. 9-12(b), 
the recombination current flowing through R x is the base current, I b . Hence, 

*» = nr = m Veb ' \ = r^ < 9 " 46 ) 

/ft I e {l — cc ) 1 - a 

Also, R 2 = ?f = i% = H (9-47) 

l c a l e a 

Finally, we see from Sec. 5-3F that 

d = C 2 = C D (9-48) 

where C D is the diffusion capacitance. For a base region of constant resis- 
tivity, Eq. (5-15) gives the diffusion capacitance of the junction to be 

eW 2 

c ° = Wtd p i * < 9 - 49 > 

where I E is the total emitter current across the junction. 



214 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 




(a) 




FIG. 9-13. (a) Resistance and capacitance values shown on the 
circuit of Fig. 9-1 2(b). (b) Replacement of components in dotted 
box in part (a) by their Norton equivalent. 



B'Q- 



C n ^ 



r<0 



a Q»V cb - 



AA/V 



/(l-ao) 



!JiC D 



B'Q- 



V„, 



1-a, 



C n ^ 



(a) 



v<B± 



A/VV 



A<(1 - «o) 



lxC L 



«0M 



6E 



(b) 



FIG. 9-14. (a) Figure 9— 13(b) redrawn in the common emitter 
connections, (b) Approximation of the current generator in the 
input circuit. 



SEC. 9-6 



THE HYBRID-77 REPRESENTATION 



215 



The values found in the preceding paragraph are shown in Fig. 9-1 3(a). 
The voltage generator and the resistance shown in the dotted box in this 
diagram can be replaced by their Norton equivalent circuit to give Fig. 
9-1 3(b). 

Figure 9-1 3(b) can be redrawn as a common emitter circuit to give 
Fig. 9-1 4(a). The input current generator will next be considered and two 
approximations will be made. In the first place, the current is proportional 
to the voltage across the CB' terminals. However, for a transistor operating 
in the common emitter connection, the voltage gain is considerably greater 
than unity, and so we can write V cb > a V ce . The value of the current generator 
now becomes « /zK ce /r e and it will load the collector and emitter terminals as 



A*(l-"o) 




6E 



B'O 



V,., 



l-«o 



Cn^ 



/u(l - ao) 



c D 



«0M 




8n,V b 



(b) 

FIG. 9-15. (a) The circuit of Fig. 9-1 4(b) with the current genera- 
tor divided into two parts, (b) The -n circuit for the intrinsic 
transistor in the common emitter connection. 

if it were a resistance of value rj(a fi) as shown in Fig. 9-14(b). Secondly, we 
note from Eq. (9-42) that [x is very small. In the input circuit, therefore, the 
generator current can be neglected in comparison with I c , which is flowing in 
a connecting lead. The two approximations are shown in Fig. 9-14(b). 

The remaining current generator in Fig. 9-1 4(b) has been divided in 
Fig. 9-1 5(a) into two generators which have a common junction at the emitter. 
The two circuits are electrically identical since a constant current generator 
may be split in this manner without altering its effect on the circuit. From 
Eq. (9-47), 



'c " eb' Sm * eb' 



(9-50) 



216 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



where g m = a /r. Thus the generator now in the input circuit supplies a 
current which is proportional to the voltage across its terminals. This 
generator can be replaced by a resistance, and, because of the direction of 
current flow, the resistance is negative and has the value 



6m ' e 



Sm 



f. 



(9-51) 



This negative resistance cancels out the existing positive resistance of the 
same magnitude already across the input terminals. The final circuit for the 
intrinsic transistor is given in Fig. 9-1 5(b). It is a tt representation of the 
approximated transmission line for a common emitter circuit. The direction 




Bo -A/W- 



r b'e — 

1250 n 



V b - e ^z 



r 6 - c =2.5M 
C 6 . c = 10pf 

C b - e = r " = 

1500 pf 50 K 




8m V Ve = 

0.039 V b . e 
amps 



-OC 



-OE 



(b) 

FIG. 9-16. (a) The hybrid-vr circuit for the common emitter con- 
nection, (b) Typical values for the hybrid-^ circuit. 

of the current generator and the polarity of the voltage to which it is related 
have both been changed to conform to current practice. 

The complete hybrid-vr circuit is shown in Fig. 9-1 6(a). Here, the three 
important components outside of the intrinsic transistor, namely r w , C je , and 
C jc , have been added. (Compare this common emitter circuit with the com- 
mon base circuit given in Fig. 9-3.) In addition, a has been replaced by 
unity where this approximation causes little error. Figure 9- 16(b) shows the 
common designations for the various elements of the circuit. Values given in 
this diagram apply to a typical medium frequency, alloy-junction, pnp 
transistor operating at I E = 1 milliamp and V CE = — 5 volts. 

The quantity, g m = a /r, has the dimensions of mhos or amps per volt. 
It is sometimes called the "intrinsic transconductance." From Sec. 4-7, 



SEC. 9-6 



THE HYBRID-77 REPRESENTATION 



217 



r € = 25.8/Tb ohms, where I E is in milliamps. Thus g m is known for a given 
emitter current and, in the typical case shown in Fig. 9-1 6(b), g m = 39 milli- 
amps per volt when I E = 1 milliamp. 

The elements of the hybrid-77 circuit are substantially independent of fre- 
quency in the usable range of the transistor. In addition, the variation in the 
values of the components of the circuit as the quiescent voltage or current 
are changed can readily be calculated. These advantages are further discussed 
in Chap. 10. 

9-7 Variation of Alpha with Emitter Current 

The value of alpha for the majority of transistors lies in the range 0.95 to 
0.995. The CE short-circuit current gain, j8, is given by p = a/(l — a) and so 



50 



40 



Si 

C H 

O <JQ 

I.S 



II 



» 20 ■' 



20 30 40 50 60 

Collector current I E in ma 



70 



80 



FIG. 9-17. Variation in h fe with emitter current for a pnp tran- 
sistor. Considerable variation in this curve is found for different 
transistors. 

the corresponding range of values of ^ is 19 to 199. More specifically, since 



or 





J8- 


a 








1 - a 




dp 




1 


1 - 


1 


da 


(1 


-af 


a 




48 


da 







p + 1 



1 - 



(9-52) 



Since alpha is close to unity, Eq. (9-52) shows that the CE current gain is 
extremely sensitive to small changes in alpha. In this section, the variation of 
alpha with quiescent emitter current is discussed. 

There are three mechanisms which account for the change in alpha. 



218 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



n-type 



1. At very low current densities: recombination in the emitter-base de- 
pletion region causes a reduction in current entering the base. 

2. At high current densities: the number of majority carriers in the base 
rises, and an electric field is produced which aids the passage of 
minority charges through the base region. This results in an increase 
in alpha. 

3. At very high current densities: the large number of carriers in the base 
increases the base conductivity and decreases both the emitter efficiency 

and the lifetime of the carriers in the 
base region. This causes a drop in 
alpha as the emitter current is increased 
(see Fig. 9-17). 

At low current densities, it is found 
that recombination in the depletion 
regions of the emitter-to-base junction 
forms a high proportion of the total 
emitter current. This is responsible for 
a small value of alpha when the emitter 
current is low. As the emitter current 
is increased, recombination processes 
capture a smaller proportion of the 
emitter charge carriers and alpha 
rises. 

Up to now, it has been assumed 
that small-current conditions applied 
to the transistor and that the majority 
carrier density was substantially con- 
stant with distance. When the emitter 
current is high, this condition cannot 
be accepted and the variation of the 
majority carrier density with distance 
in the base region must be considered. 
Figure 9-18 shows the majority and 
minority carrier density in the base 
region under large-current operation. 

This carrier density profile is obtained by the condition for charge neutrality 

in the base region 




o w 

Distance in active base region — *■ 

FIG. 9-18. Carrier density in the base 
region when the emitter current density 
is high. 



and, assuming N a ~ 0, 



n + N a = p + N d 



P = N d 



(9-53) 



(9-54) 



Under small-current conditions, p is assumed to be small everywhere, giving 
n = N d (a constant), as shown previously. Under large-current operation, 



SEC. 9-7 VARIATION OF ALPHA WITH EMITTER CURRENT 219 

the value of p at x = no longer can be neglected in comparison with N d , 
and so Eq. (9-54) applies as shown in Fig. 9-18. 

Charge neutrality is not exactly obeyed in the base region. The conditions 
shown in Fig. 9-18 lead to a diffusion current formed by electrons moving 
across the base region from left to right because of the concentration gradient. 
Since the emitter efficiency is close to unity, the net electron current is ap- 
proximately zero and some mechanism must be operating to cause an equal 
current in the opposite direction. This mechanism is that of drift of electrons 
in an electric field. The field is produced by the diffusion of electrons which 
produces a small imbalance of the charge neutrality condition and produces 
a field £. The difference between the positive and negative charge density 
distributions is very small, and Eq. (9-53) can still be used in the analysis 
except where the electric field is to be calculated. 

Since the total electron current is zero and both drift and diffusion take 
place, 

/- = enpj + eD n ^ = 

or g = -Ejl^L (9-55) 

n[i n dx 

Since dn/dx is negative, the field is from left to right. Assuming that the de- 
parture from charge neutrality is small, we may differentiate Eq. (9-54) with 
respect to x and write 

^ = * (9-56) 

dx dx 

From the Einstein equation of Sec. 4-8, 















(9-57) 


and so 




nfjip dx 








(9-58) 


This electric field also affects the hole current which is 


flowing, 


since 






/ + 


= epii p £ - eD p -£ 


dp 
'dx 






(9-59) 



Under small-current conditions, p « n and the current flow is wholly by 
diffusion. As/? increases, part of the total current is carried by drift and, from 



220 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

Eq. (9-59), this can be considered as an apparent increase in the diffusion 
constant, D p . In the limit, p -> n, and D p apparently doubles. The field is 
responsible for aiding the holes in their passage across the base and less re- 
combination takes place since they spend less time in the rc-type base region. 
This effect is responsible for an increase in alpha as the emitter current is 
raised. 

In Sec. 5-3 B, the emitter efficiency was given as 

y-i-?£ (9-60) 

where a b and o e are the conductivities of the base and emitter regions re- 
spectively. Under large-current conditions, there are many charge carriers 
present in the base region as shown in Fig. 9-18. Thus the base conductivity, 
o- b , increases and the emitter efficiency falls. The lifetime for holes in the base 
region, r p , is also decreased by the presence of a large number of electrons and 
this reduces the transport factor, j8j. As the emitter current increases to very 
large values, these effects take control, and alpha decreases. 



9-8 The Drift Transistor 

So far in this book, the transistor has been assumed to have a base region of 
constant resistivity. Under small-current conditions, it has been shown that 
the injected minority carriers move across the base by diffusion and that no 
appreciable electric field exists in the base. For this reason, we may use the 
term "diffusion transistor" for the device although recognizing, from 
Sec. 9-7, that an electric field will be present when the emitter current density 
is high. 

In the diffusion transistor, the transit time for minority carriers across the 
base forms an upper limit on the frequency of operation of the device. Re- 
ducing the thickness of the base decreases the transit time and increases the 
usable frequency range of operation, but if the base is made too thin, other 
limitations arise. For example, the base spreading resistance increases as the 
base thickness is reduced and, whatever changes in thickness, conductivity, 
and area of the regions are made, a point is reached where it is uneconomical 
to seek a higher frequency of operation for the diffusion transistor. 

The drift transistor overcomes some of the limitations of the diffusion 
transistor and can be operated at frequencies up to a few thousand mega- 
cycles per second. Devices are commercially available which give substantial 
gain in a common emitter circuit at several hundred megacycles per second. 
This is accomplished by reducing the transit time of the minority carriers in 
the base region and by ensuring that the collector capacitance is small. 

The drift transistor is manufactured with a nonuniform base conductivity. 
As shown below, this results in a "built-in" electric field which aids the 



SEC. 9-8 



THE DRIFT TRANSISTOR 



221 



Near-intrinsic 
material 



PT 



r 



minority carrier motion across the base region. Both drift and diffusion occur 
in the drift transistor, and the combination of the two current-carrying 
mechanisms reduces the transit time and so increases the upper frequency 
response. In a pnp transistor of this type, the /7-type impurity is diffused into 
the base region under very carefully controlled conditions of temperature and 
pressure so that an impurity gradient is produced in the base region. In many 
cases, the impurity density in the base can be assumed to drop approximately 
exponentially from the emitter to the collector junction and this results in a 
constant "built-in" field. There are many methods of producing a drift 
transistor; all of them involve the diffusion of an n- or/?-type impurity from a 
vapor into the base region of a transistor. The manufacturers' literature should 
be consulted for the method used for any 
particular type (see also Sec. 5-6). 

The way in which the impurity den- 
sity varies with distance in a typical 
drift transistor is shown in Fig. 9-19. 
Close to the collector junction, the base 
resistivity is high and the material is 
near-intrinsic. The charge depletion 
region at the collector end of the 
junction extends well into the base as 
shown, and this results in a low collec- 
tor-to-base junction capacitance [Eq. 
(3-24)]. This is an important character- 
istic of the drift transistor and is partly 
responsible for its good high frequency 
performance. 

The electric field which is present by virtue of a nonuniform base con- 
ductivity can be calculated by considering the majority charge distribution in 
the base. ¥ ox a. pnp drift transistor, the majority carrier density has the profile 
shown in Fig. 9-19, and so electrons will diffuse in the direction of decreasing 
concentration, that is, from left to right. As previously discussed in Sec. 9-7, 
this slightly unbalances the condition of charge neutrality and results in an 
electric field, *f , which produces a drift of electrons in the opposite direction. 
Writing the equations for the total current density of electrons in the base 
region, we see that 

, _ _ dn 

J- = e\L n ni 4- eD n ^- 



Exponential 
doping-^ 



Collector- 
to-base 

depletion 
region 



Fl G. 9- 1 9. Structure of drift transistor. 







(9-61) 



since no net majority carrier flow occurs in the base. Thus, 



D n dn 
\h n n dx 



(9-62) 



222 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

From the Einstein relation of Sec. 4-8, 

^ = — (9-63) 

and so 

# = _kTdn 
en ax 

If $ is to be made constant throughout the base, Eq. (9-64) can be integrated 
to give 

n = n txp^-~^x ) j (9-65) 

where n = n at x = 0. Furthermore, if n = n w at x = W, we can solve 
Eq. (9-65) for <f to give 

'-^(Si) (9 - 66) 

This is the constant "built-in" field that is produced by the exponential 
majority carrier density distribution. It has a positive value since n > n w 
and is acting in the direction from emitter to collector. For the majority car- 
riers (electrons), as we have seen, the currents attributable to drift and 
diffusion are equal and opposite and the net current is zero. For minority 
carriers (positive holes), the drift and diffusion currents add and the total hole 
current density is 

J* = e M * - eD p f x (9-67) 

where dpjdx is a negative quantity. The total hole current density across the 
base, 7 + , is a constant. Therefore, the first order differential equation (9-67) 
can be solved to give 

p -^r t { 1 ~ ^Y x ~ ^W\ (9 " 68) 

where the constant of integration has been evaluated by noting that p = at 
x = W. From Eqs. (9-64) and (9-66) 

At the beginning of the base region, where x « W, 



Thus, 



P = 



SEC. 9-8 THE DRIFT TRANSISTOR 223 

if n w /n « 1. In practice, the doping is such that n w /n is of the order of 1/8. 
Thus, nearly 90 per cent of the current is carried by drift at the beginning of 
the base region but as x -» W, diffusion increases. The transit time of the 
holes across the base is much reduced by the presence of the "built-in" field, 
and so the transit time is reduced. 

The term " diffused-base transistor" has come into general use in the 
semiconductor device industry. The process of diffusion, which is now 
standard in the production of high frequency units, results in a base region 
with a nonuniform impurity density (approximately exponential). The an- 
alysis of the drift transistor applies to some extent to all diffused-base 
transistor constructions. Whenever the base conductivity is nonuniform, an 
internal electric field will be developed in the base region. Diffused-base 
transistors are now designed for particular applications. By varying the base 
width, conductivity values, junction areas, etc., the important properties of 
the transistor which determine its high frequency and pulse characteristics can 
be suited to the application (see Chap. 10). 

9-9 Transistor Noise 

The random noise generated in a device determines its ultimate sensitivity as 
an amplifier. There are three sources of noise in a transistor: resistance noise 
caused by thermal agitation ; shot noise caused by the discrete nature of the 
charge carriers ; and modulation or semiconductor noise. 

Resistance noise is due to the random motion of the holes and electrons 
in a material and can be represented by a mean square noise voltage of the 

1? = AkTR Af (9-72) 

form where k is Boltzmann's constant, T is in °K, R is the resistance of the 
material and Af is the band of frequencies over which the noise is measured. 
This is known as "white" noise since it has a constant frequency spectrum, 
that is, for a constant bandwidth, Af, v 2 does not depend on frequency. 

Shot noise is produced wherever a current is flowing in the transistor, and 
it also is "white" noise. The current is carried by individual carriers (holes 
or electrons) and so it fluctuates slightly about its mean value. The mean 
square fluctuation in the current is 

? = lelAf (9-73) 

where /is the average current. If this mean square current is flowing through 
a resistance, R, it will produce a mean square voltage across the terminals of 

v 2 = 2eIR 2 Af (9-74) 

Modulation or semiconductor noise is also known as flicker noise. It is 
entirely a low frequency phenomenon and is of considerable importance in 



224 



PHYSICAL CHARACTERISTICS OF THE TRANSISTOR 



CHAP. 9 



low frequency and d-c amplifiers. Its origin is thought to lie in the semicon- 
ductor crystal imperfections and in surface effects. It is not "white" noise, 
but can be represented by a mean square voltage of the form 

~s AI'Af 



f 



(9-75) 



where A is a constant and / is the current flowing. Thus, the mean square 
voltage is inversely proportional to the frequency, /. This type of noise is 
predominant below about 1 kc/sec in most transistors. 

The transistor adds a certain amount of noise power to any signal that it 
amplifies. However, the input signal to the transistor includes thermal noise 
of its own, generated by the source resistance, R s . An assessment of the 



10 



Region 1 



Region 3 



C 



Region 2 



0.01 0.1 1 10 100 

Frequency in kc/sec — *■ 



1,000 



FIG. 9-20. Noise figure of a transistor as a function of frequency. 
Source resistance, R s = 500 ohms. 

transistor's noise performance can be obtained by finding the fractional in- 
crease in noise caused by the transistor. The noise figure, F, is defined to be 
the ratio of the total noise output power to the noise output power attribut- 
able solely to the thermal noise in the source. Hence, 

Ptn 



ApPsN 



1 + 



AP, 



(9-76) 



where P TN is the internally generated transistor noise power, P SN is the thermal 
noise in the signal, and A p is the transistor power gain. The value of P SN per 
unit bandwidth does not vary with frequency as shown by Eq. (9-72), assum- 
ing a resistive source. The power gain, A p , and the transistor noise, P TN , are 
frequency dependent, and thus the noise figure of the transistor will vary with 
frequency. 



SEC. 9-9 TRANSISTOR NOISE 225 

The noise figure is usually quoted in decibels where 

F«-101og, (l +J7T;) (9-77) 

Figure 9-20 shows a curve of the noise figure versus frequency for a typical 
transistor. Between about 1 kc/sec and 100 kc/sec, the noise figure is 6 db. 
From Eq. (9-77), this means that the total noise power at the output terminals, 
under these conditions, is composed of one part determined by the external 
noise source (R s ) and three parts generated in the transistor. Noise figures as 
low as 1 db can be obtained using selected transistors under ideal conditions. 

The three regions of Fig. 9-20 can be identified from Eqs. (9-72) through 
(9-77). In region 1, the modulation noise is high in the transistor, and the 
noise figure increases as the frequency decreases. In region 2, modulation 
noise can be ignored, and thermal and shot noise predominate. The noise 
figure increases in region 3 mainly because the power gain of the transistor 
decreases as the frequency is raised. The demarcation frequency between 
regions 2 and 3 is approximately the geometrical mean of the alpha and beta 
cutoff frequencies. The "spot" noise figure given by many manufacturers is 
measured at 1 kc/sec. 

Figure 9-20 has been drawn for a source resistance of 500 ohms. For 
most transistors, the noise figure is found to be at its lowest value when R s is 
between 100 and 1000 ohms. For low noise operation, the transistor should 
be operated with emitter current and collector-to-base voltage in the low-to- 
medium range. 

9-10 Surface Effects in Transistors 

Transistors are very sensitive to the conditions existing on the semiconductor 
surfaces, and surface contamination in device manufacture is a major 
problem. In the early days of transistors, surface contamination caused a 
slow deterioration of the transistor characteristics. This problem has been 
largely overcome by extensive surface treatment and elaborate encapsulation 
of the device, but the surface properties still play some part in determining 
the characteristics of the transistor. 

If the surface of a semiconducting crystal is rough, there will be many re- 
combination centers present and the lifetimes of the charge carriers will be 
reduced (see Sec. 1-20). Since the lifetimes in the bulk material are unchanged, 
the hole and electron densities close to the surface are reduced and there will 
be a flow of carriers towards the surface. The net current flow is zero since 
both types of carrier are involved. Conditions at the surface are charac- 
terized by the surface recombination velocity, s, defined by 

e(Ps ~ Po) 
where p s and p are the thermal equilibrium hole densities at the surface and 
in the bulk material respectively, and J s is the hole current density flowing to 



226 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

the surface. Values of s vary from about 2000 cm sec -1 (for a sand-blasted 
surface) to less than 200 cm sec -1 (for an electrolytically etched surface). In 
the base region of a pnp transistor, any positive holes which travel to the 
surface and recombine are lost from the collector current. This loss maybe 
reduced by making the surface recombination velocity small and by ensuring 
that the collector subtends a large solid angle to every point on the emitter. 
The diffused-base transistor is good in this respect since the drift field aids 
the minority carriers to pass directly from the emitter to the collector. 

Many substances which are present in the atmosphere, e.g., water, form 
a surface layer on germanium or silicon which is only a few molecules thick. 
This layer readily becomes charged and produces a high electric field at the 
surface of the semiconductor and alters the conditions in the device. Manu- 
facturing processes are now being used which "passivate" the surface to 
reduce the effects of surface films and so prevent the variation of transistor 
characteristics with time. 

The collector cutoff current, I CB o, is markedly dependent on the surface 
conditions of the semiconducting material. The wide variation in I CBO 
between transistors with apparently the same construction is often the result 
of differences in surface conditions. 

9-1 1 Punch-through 

In Sees. 5-3C and 9-5 base width modulation in a diffusion transistor was 
discussed. It was shown that a change in collector-to-base voltage, V CB , is 
responsible for a change in the active width of the base region, W. The 
width of the depletion region at the collector end of the base is proportional 
to Vc B , where n is approximately — \ for the alloy-junction diffusion transistor, 
and — J for the diffused-base drift transistor. When a high reverse collector-to- 
base voltage is applied to the transistor, the depletion width may be so large 
that the active base region disappears entirely and the emitter-to-base and 
collector-to-base depletion regions merge. Transistor action is no longer 
possible under these conditions. This phenomenon is known as punch- 
through and the particular collector-to-base voltage at which it occurs is the 
punch-through voltage. 

Punch-through is one of the conditions which determine the upper limit 
of allowable collector-to-base voltage. This voltage will be specified by the 
manufacturer. Punch-through is not necessarily harmful to the transistor. 
However, if the resistance in the external circuit permits an excessively high 
current to flow under punch-through conditions, the transistor will be 
damaged. 

9-12 Epitaxial Films 

Starting in 1960, several manufacturers added the epitaxial process to the pro- 
duction of diffused-base transistors. In transistor parlance the term epitaxial 



SEC. 9-12 EPITAXIAL FILMS 227 

refers to the deposition of a thin, highly resistive layer of single crystal semi- 
conducting material on a low resistivity substrate where the crystal orienta- 
tion of the upper layer is the same as that of the substrate. The principal use 
of epitaxial films has been to improve the collector saturation voltage and 
speed of diffused-base transistors. 

The epitaxial film is deposited by a vapor process. The semiconductor 
material in the form of a compound is vaporized and a chemical or thermal 
decomposition is made to occur in the vicinity of the single crystal substrate. 
The substrate is heated and semiconductor atoms from the vapor deposit 
epitaxially on the substrate surface. By adding compounds of doping elements 
to the vapor stream, deposition of the doping element also occurs, and an 
epitaxial film of the desired resistivity and thickness is grown on the substrate. 

In the conventional mesa transistor (see Fig. 5-15), the collector material 
is of fairly high resistivity to avoid collector-to-base breakdown at low 
voltages. However, this means that the ohmic resistance of the collector 



Emitter contact . Base contact 

^71 1 mSL 



Epitaxial collector region 

ssistivity s 
(collector) 



J*- Low resistivity substrate 



V 



Ohmic contact 



FIG. 9-21. Epitaxial mesa diffused-base structure. Compare 
with Fig. 5-15. 

region is appreciable and the collector-to-emitter saturation voltage is corres- 
pondingly high. In addition, the high resistivity material has a long lifetime 
and minority charges will, therefore, be stored in the collector region for a 
comparatively long time and over a large distance. This causes a limit to be 
set on the switching time of the transistor. 

The epitaxial mesa transistor is shown in Fig. 9-21. It employs a collector 
wafer of low resistivity, on top of which is deposited the epitaxial high re- 
sistance layer as shown in the diagram. The reduced resistance of the col- 
lector region ensures that the collector-to-emitter saturation voltage is small 
and it may be as low as one tenth of its value for the conventional structure 
(e.g., down to 0.4 volts for I c = 100 ma). The lifetime of minority carriers in 
the high resistivity epitaxial layer is much smaller and so switching times are 
reduced. Other advantages are lower collector-to-emitter capacitance and the 
retention of high current gain at high current levels. 

BIBLIOGRAPHY 

Cote, Alfred J., Jr., and J. Barry Oakes, Linear Vacuum-tube and Transistor 
Circuits, New York: McGraw-Hill Book Company, Inc., 1961 



228 PHYSICAL CHARACTERISTICS OF THE TRANSISTOR CHAP. 9 

DeWitt, David, and Arthur L. Rossoff, Transistor Electronics, New York: 
McGraw-Hill Book Company, Inc., 1957 

Gartner, Wolfgang W., Transistors: Principles, Design and Applications, 
Princeton, N.J.: D. Van Nostrand Company, Inc., 1960 

Middlebrook, R. D., An Introduction to Junction Transistor Theory, New 
York: John Wiley & Sons, Inc., 1957 

Pettit, Joseph Mayo, and Malcolm Myers McWhorter, Electronic Amplifier 
Circuits, Theory and Design, New York: McGraw-Hill Book Company, 
Inc., 1961 

Wolfendale, E., ed., The Junction Transistor and its Applications, London: 
Hey wood & Company Ltd., 1958 

PROBLEMS 

9-1 Explain the meaning of the base transport factor, j8*. Find j8* as a 
function of frequency and calculate the frequency at which fi* drops to 
90 per cent of its low frequency value if WjL p = 0.0785 and r p = 60 
jusec. 

ans. 870 kc/sec 

9-2 Given the hybrid-77- circuit of Fig. 9-1 6(b), find the value of the four h 
parameters. 

9-3 Explain in words, equations, and diagrams the phenomenon of base 
width modulation. 

9-4 A pnp germanium transistor has abrupt junctions and is operated at 
room temperature. The following details of the transistor are known. 
Emitter: p p = 0.07 ohm cm; Base: p n = 1.6 ohm cm; Collector: 
P P = 0.40 ohm cm; Metallurgical base width = 2 x 10~ 3 cm;a =0.98 
(assumed equal to /?*); Collector-to-base area = 5 x 10" 3 cm 2 ; 
V CB = 10 volts (reverse bias); I E = 1 ma. Determine the following 
quantities, (a) contact potential between collector and base, (b) collec- 
tor-to-base depletion width, (c) r b > ei (d) jjl from Eq. (9-42). 

ans. 0.268 volts, 4.16 x 10~ 4 cm, 1290 ohms, 3.25 x 10~ 4 

9-5 For the transistor of Prob. 9-4, calculate (a) C b > e with the condition 
C D > C je , (b) r b , c , (c) C b . c taking /xC D into account, (d) r ce , (e) g m . 
Draw the hybrid-77 circuit for the transistor showing the numerical 
values for all the components. 



10 



High 

Frequency and 

Pulse Operation 

of the 

Transistor 



The hybrid-7r equivalent circuit, which was derived 
in Chap. 9, is used here to determine the upper 
frequency response of the transistor stage. The 
analysis is limited to the common emitter con- 
figuration and includes a calculation of the gain and 
bandwidth for an iterative stage. The hybrid-vr 
model is then compared with the actual transistor, 
and it is shown that a simple, empirical modifica- 
tion of the circuit leads to a more exact representa- 
tion at high frequencies. 



230 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



The hybrid-77 circuit is also used initially in the analysis of pulsed opera- 
tion of the transistor. However, the concept of charge control is briefly de- 
veloped and it is shown that this new approach offers advantages in analyzing 
and understanding the transient conditions of switching. 

10-1 The Hybrid-K Circuit 

Figure 10-1 shows the hybrid-77 model of the transistor that will be used in 
the first part of this chapter. It is an active tt circuit with the base spreading 



£0- 



^VW 



125012 



EO- 



1500 pf v 



r b ,=2.5M 

Q, c =i0pf 



50 K 




-oC 



g m v b - e = 

0.039 V b . e 
amps 



■OE 



FIG. 10-1. The hybrid-7r circuit with typical values. (I E = 1 ma). 

resistance, r w , added to the input lead. This circuit is developed from the 
transmission-line analog of the transistor in Sec. 9-6. 

The important elements of this circuit are identified below. 

r bb >, the base spreading resistance, is the ohmic resistance between the base 
y lead and the active base region. It is substantially independent of operating 
conditions. 

r b > e is the a-c resistance between the active base, B', and the emitter terminal, 
E. Its value is given by rj{\ — a ) where r € — kTleI E = 25.8// £ mhos at 
300°K where I E is in milliamps. In the region where a can be considered 
a constant, r Ve is inversely proportional to I E . 

C b > e is the sum of the emitter junction capacitance, C je , and the emitter-to-base 
diffusion capacitance, C D . In many cases C D predominates. From Eq. (5-15), 
C D is proportional to I E . 

C b > c is the sum of the collector junction capacitance, C jc , and a small fraction 
of the diffusion capacitance, C D . C jc usually predominates and is propor- 
tional to Vqb where n is between — \ and — \ depending on the type of 
transistor (see Sec. 3-4). 

g m is the transconductance and is of magnitude a /r e . It is, therefore, approxi- 
mately proportional to I E and has the value of 0.039 mhos when I E = 1 
ma at 300°K. 

All of these values should be considered approximate and considerable 
deviations will be found in practice. To the first order of accuracy, all the 



SEC. 1 0-1 



THE HYBRID-77 CIRCUIT 



23 



elements are independent of frequency, and the change in the major com- 
ponents of the circuit can be calculated as I E or V CE is changed. These two 
factors provide major advantages over other transistor representations which 
might be used. 

An analysis of the CE stage at high frequencies will be carried out using 
the hybrid-77- circuit. This gives an adequate representation for most transis- 
tors at frequencies an order of magnitude less than the alpha-cutoff frequency. 
For higher frequencies, modifications must be made to the circuit if it is to be 
used with accuracy. These modifications are briefly discussed in Sec. 10-6. 

10-2 The Miller Effect 

Figure 10-2(a) shows the a-c circuit of a transistor amplifier in the CE con- 
nection with a load R L and a constant current input source. Figure 10-2(b) 



/, <> 



' bb- 



(a) 
7 3, C b >c 



\h 



Vu; 



\g m Vb 



(b) 

FIG. 10-2. (a) The a-c circuit of a CE amplifier, (b) Simplified 
hybrid-^ equivalent. 



shows a simplified hybrid-77- equivalent. Two components have been omitted 
from the complete hybrid-?! circuit of Fig. 10-1. These are: r ce , which is 
significantly larger than the load resistance, and r b > c , which is usually so high 
that its effect can be neglected. Typical values for these components are given 
in Fig. 10-1. 

This section will consider the simplified hybrid-rr circuit and show that the 
effective capacitance between B' and E is increased by the presence of C b > c , 
and that a new equivalent circuit can be drawn (Fig. 10-3). This is analogous 
to the Miller effect in vacuum tube circuitry. 



232 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

From Fig. 10-2(b), 



and 
Furthermore, 

In practice, I 3 « I and so 



h = h + h 

h = j<*>C b > e V b > e 

h = {V b > e - V )ja>C b 

K = -gmV b >eR L 



Therefore, 



and 



h =]«>c b ,y b . e + ja>c b , c (v b , e - v ) 

= ]<»V v iP v . + (1 +g m R L )C b , c ] 

-u- =i*e + (l +g m Ri)C b . c ] 



(10-1) 
(10-2) 

(10-3) 

(10-4) 

(10-5) 
(10-6) 



Thus, as far as the input side of the transistor is concerned, the effect of the 
feedback current, 7 3 , is to increase the capacitance to the value C, where 

C = Q e + (1 + g m R L )C b . c (10-7) 

This effect was first noted by J. M. Miller for the case of vacuum tubes in 1919. 



/,- 



'66' 






SmVb'e 



FIG. 10-3. Simplified equivalent circuit showing effective capacitance, C. 

When the collector load, R L , is so small that the output circuit is unaffected 
by the feedback current, I 3 , the equivalent circuit can be drawn as in Fig. 10-3. 
This is the usual case for an iterative resistance-capacitance coupled stage. 

Using the values given in Sec. 10-1 and Fig. 10-1, with an emitter direct 
current of 1 ma, and with R L = 1000 ohms, 

C = 1500 + (1 + 39)10 

= 1900pf (10-8) 

This capacitance is in parallel with r b > e and the two elements form a CR 
circuit which will be shown to limit the frequency response of the transistor 
stage. 



SEC. 10-3 FREQUENCY RESPONSE OF THE TRANSISTOR: HYBRID-77 MODEL 233 

10-3 Frequency Response of the Transistor 

in the Common Emitter Connection: Hybrid-n Model 

From Fig. 10-3, the impedance of the parallel resistance-capacitance com- 
bination, r b > e C. at an angular frequency a> is 

Z = ~ ^ — - = *-p (10-9) 

The current gain of the transistor in the common emitter connection is 

*i = T= gmVve #- = r+TTr (1(M0) 

from Eq. (10-9). Now, g m = a jr e , and r Ve — rj(l - a ), hence, 

^i = T-^-l r ?,=■! h—^ (10-11) 

At low frequencies, Eq. (10-11) shows that A t = j8 . Recalling that 

C = C h . e + (1 + g m R L )C b , c 

Eq. (10-11) gives the frequency response of the transistor in the CE con- 
nection, that is, 

' 1 + jwr b , e [C b - e + (1 + g m R L )C b . c ] y ' 

Under short-circuit conditions with R L -» 0, 

a - i!° (10-13") 

At an angular frequency, c* 0i where 

». = 2 ^* = r re '+ c i ~ rV (1(M4) 

the current gain of the stage will have fallen to 1/V2 of its low frequency 
value. The short-circuit current gain as a function of angular frequency can 
be written as 

«-> = r+fea " r+fcro (1 °- 15) 

Figure 10-4 shows a plot of the magnitude of the short-circuit current gain 
versus frequency for a hybrid-77- circuit. The current gain in the figure is 
given in decibels (db) where the number of db = 20 log 10 (hlh)- At low 
frequencies, p = 50 corresponds to 34 db of gain. At the frequency /^ the 
gain has fallen to ft/V2 which corresponds to 31 db or a loss of 3 db. The 
frequency where there is a 3 db loss of gain is a measure of the useful 



234 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



band-width of the transistor when used in a stage with a low frequency gain 
of j8 . It will be shown in the next section that this does not limit the operation 
of the common emitter stage to frequencies lower than f fi9 provided that a 
lower value of gain at low frequencies can be tolerated. f is known as the 
beta-cutoff frequency of the transistor. 

At frequencies considerably in excess of/^, the imaginary term in Eq. (10- 
15) becomes much greater than unity. The drop in gain for one octave in- 
crease in frequency is then 6 db, and the curve shown in Fig. 10-4 becomes a 



(31 db) 




FIG. 10-4. High frequency response of a typical hybrid-^ circuit 
showing f e and f T . j8 is assumed to be 50. 

straight line. The frequency where \A t \ is unity is named f T , and can be 
obtained from Eq. (10-13) by noting that co T r b > e (C b > e + C b > e ) » 1. Thus, 

£o _ Po 



= 2tt/ t = 



r b > 



(10-16) 



i\Cb'e + Cb'c) r b'eCb' e 

From Eqs. (10-14) and (10-16), 

Pof,=fr 00-17) 

for the hybrid-7r circuit, and so f T is the gain-bandwidth product for the 
transistor in the CE connection. f T is also a useful parameter in pulse opera- 
tion of the transistor and it will be discussed further in Sees. 10-6 and 10-10. 
The calculations shown above, starting with Eq. (10-13), apply only 
when the output terminals are short-circuited to alternating currents. If a 
high collector load is inserted so that the capacitance C [Eq. (10-7)], includes 
an appreciable contribution due to the term g m R L C b > c , then the angular 
frequency corresponding to a loss in gain of 3 db will occur at 

1 

" 3db r b , e [C b , e + (l +g m R L )C b . c ] 



SEC. 10-3 



FREQUENCY RESPONSE OF THE TRANSISTOR: HYBRID-tt MODEL 



235 



from Eq. (10-12). Assuming g m R L » 1 

1 



1 



'3 db 



r b'eCb' e + fb'eSm^-LCb'c r b'e^b'e + Po^b'c^L 

when the values for r h > e and g m are substituted from Sec. 10-1. The 3 db 
frequency is, therefore, shown to include a time constant term, p C b > c R L , in the 
denominator. C b > c is the measured output capacitance in the CB configura- 
tion and is commonly designated C ob in manufacturers' data sheets. The out- 
put capacitance in the CE connection (sometimes designated C oe ) is p C ob and 
the resultant output time constant C oe R L = fi Q C b > c R L appears in the denomi- 
nator of the equation. To obtain maximum bandwidth from a transistor 
amplifier, R L is chosen so that fi Q C b - c R L « r b . e C b - e . Note that reducing R L 
slightly increases the current gain of a stage but materially reduces the 
voltage gain. In a multistage amplifier, it is the current gain which is of 
importance and R L is usually small enough that the inequality is satisfied. 

10-4 Frequency Response of the Iterative 

Common Emitter Stage: Hybrid-n Model 

Figure 10-5 shows the equivalent circuit of a transistor stage where the 
resistance R s shunts the input current generator. In a multistage transistor 




FIG. 10-5. Equivalent circuit of typical CE iterative stage. 

amplifier, R s will be composed of the load resistance of the previous stage in 
parallel with resistances in the interstage coupling network. The presence of 
R s has an effect on the frequency response of the stage as shown below. 
If I b is the alternating current entering the base terminal of the transistor, 

h = Rs_ 



(10-18) 



(10-19) 



Z + r bb . + R s 
where Z is given in Eq. (10-9). The voltage across Z is V b > e , hence, 

Vb'e _ yh _ ZR S 

I t I t Z + r bb , + R s 

Substituting for Z from Eq. (10-9), and putting R t = R s + r b , e + r bb >, we 
have 

VjSe = Rsfo'eKl + M b . e C)] 

Ii r b , e j(\ +jur b , e C) + r bb , + R s 

_ Rsfb'e 

Rt + J«>r b . e C(R s + r bb ) 



(10-20) 



236 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

The current gain of the stage is 

a '_o_ gm j_ b'e gm^-S'b'e / 1 rv_o i \ 

A '-I l - I, ~ R t + j<or b , e C(R s + r bb .) UU_/1J 

The current gain of the stage is 3 db down from its low frequency value at an 
angular frequency, co 3 db , given by 

Rs "F ?bb' r b'e^ Rs + r bb' 

The low frequency gain from Eq. (10-21) is 

A i0 = ^£^ (10-23) 

Rt 

The product of the low frequency gain and the angular frequency is given by 

gmRsfb'e R t 1 



(10-24) 



Rs + r w C 

If R s » r bb /, ^i co 3db = g m /C and this value can be used as a. figure of merit 
for the transistor. 

A consideration of Eqs. (10-22) and (10-23) shows that the transistor can 
be used at frequencies in excess of the beta-cutoff frequency provided that a 
lower current gain can be tolerated. For the transistor of Fig. 10-1, 

f * = ( t = 2^~C = 277(1250)1900 = 67 ° kc / sec 

If R s = oo, the low frequency gain, A i0 = g m r b > e = 49 and ct» 3db = o) . 
When R s = 1000 ohms, 

0.039(1000)1250 



2320 
and 



21 



w 3dh 2320 



/ 3db = -$£ • = Joto 670,000 = 1.44 Mc/sec 

If i? s is further reduced, the stage gain decreases and the 3 db frequency 
increases. Note that the stage gain includes the shunting effect of R s . 



10-5 Frequency Response of the Transistor 

in the Common Base Connection: Simple Model 

The calculations which have been made on the CE hybrid-77 representation 
of the transistor suggest a simple means of comparing the frequency response 



SEC. 10-5 FREQUENCY RESPONSE OF THE TRANSISTOR: SIMPLE MODEL 



237 




B'i> 



-A/W 



oC 




/, 




ao-"^c6' 



J"(l-»o). 



^r-iiCu Vcb- 



(b) 

FIG. 10-6. (a) Transmission line analog of base region, (b) A 
simplified representation of the transmission line analog. 

of the CB and CE connections. In Fig. 10-6(a) we repeat the transmission 
line analog of the base region of the intrinsic transistor previously given in 
Fig. 9-11, and Fig. 10-6(b) shows the simplified representation given before 
in Fig. 9- 13(b). This circuit can be used to find the frequency response of the 
transistor in the CB connection. 

Figure 10-7 has been drawn for the intrinsic transistor operated under 
short-circuit output conditions. When the emitter-to-base junction capacit- 
ance is included, the input capacitance, C b > e , is given by 



(-'b'e — ^D i (-j 



(10-25) 



where C je is the junction capacitance from emitter to base. The combination 
of the two resistances, rj(\ - a ) and rja , in parallel is r €i and so 



1 + jwC b > e r e 



= 1 ^ 

a 



l-a ( 



^C T 



QC 



Short 
circuit 



OB' 



FIG. 10-7. The intrinsic transistor operating in the common 
emitter connection with short circuited output. Emitter-to-base 
junction capacitance shown with dashed line. 







top = 


1 - «, 


3 ~ "cc 


the hybrid-77 


circuit, 


therefore, 










J a £ 


l f ~ 


Pofp 



238 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

and the current gain in the short-circuited common emitter connection is 

A i = a(co) = If = a ° (10-26) 

I t 1 +jcoC b > e r e 

The alpha-cutoff frequency, f a , is defined as the frequency where A t has 
dropped to 1/V2 of its low frequency value. Thus, 

oj a = 2nf a = -±- (10-27) 

from Eq. (10-26) and that equation can be written as 

«("> = 1 4. W I \ (10 " 28) 

1 +J{a>/a) a ) 

By comparison with the value for the beta-cutoff angular frequency [Eq. 
(10-14)], and noting that r b < e = rj(l — a ), 

(10-29) 



(10-30) 



10-6 Comparison of the Hybrid-n 

Circuit with the Actual Transistor 

The hybrid-77 circuit is a simplified representation of the transistor which 
serves as a starting point for electronic circuit design. When the transistor is 
operated at high frequencies or under pulse conditions, however, the hybrid-n- 
circuit provides only an approximate model for the analysis of transistor 
behavior. Fortunately, this model can be improved with some simple 
modifications, and it is then adequate for most types of circuit design. 

When measurements of transistor parameters are attempted, the experi- 
menter becomes aware of a considerable spread in the quantities that he is 
trying to measure. Variations in materials and manufacturing techniques are 
responsible for most of these deviations, and the circuit designer must 
account for the resulting spread in the characteristics. Thus, electronic 
circuit components are seldom specified to an accuracy of better than 
10 per cent, and the designer endeavors to produce a circuit which will 
operate even when all the parameters and conditions take their most un- 
favorable values. This is known as the "worst-case" design philosophy and 
is important in the design of large-scale electronic equipment where the 
failure of any circuit must be avoided. It is for this reason that circuit 
electronics is not an exact subject and, consequently, a model of a solid state 



SEC. 10-6 



COMPARISON OF THE HYBRID-tt CIRCUIT 



239 



or vacuum device is usually adequate if it represents the device to within an 
accuracy of about 10 per cent within the frequency band considered. 
The alpha-cutoff angular frequency has been given by Eq. (10-27) as 



We can substitute for C D and r € from Eqs. (5-15) and (4-45), to give 

_ 2kTD p eI E 2D P 
Wa eW 2 I F kT W 2 



(10-31) 



The hybrid-77- circuit was derived by simplifying the transmission line equi- 
valent to the transistor. When the a-c continuity equation (9-21) is solved, 
the exact relationship for the diffusion transistor is 



». = 2.43 ^§ 



(10-32) 



Therefore, the hybrids circuit shows an error of about 20 per cent in its 
prediction of the alpha-cutoff frequency from the physical constants of the 
intrinsic (diffusion) transistor. 



100 

<U 
0> 

g> 80 

G 

<o 60 

a 

CS 

£ 40 



20 



Diffusion 


/ 

/ 
/ 
/ 
/ 
/ 
/ 
/ 

transistor-^ / 

/ S 

S^r ^-Simple model 


- 




>''/' of Fig. 10-7 

L 
.11 1 1 . ■ 1 



0.1 



0.2 



0.4 0.6 0.8 1 



fife 



FIG. 10-8. Phase difference between output and input current in 
the common base connection as a function of f/f a for a diffusion 
transistor and for the model of Fig. 10-7. 

The measured phase between the output and the input current for a 
diffusion transistor also varies considerably from that predicted by the simple 
model. This is shown in Fig. 10-8 where the diffusion transistor and the 
hybrid-77- model are compared. The difference in phase angle between the 
two curves is approximately proportional to fjf a . Empirically, we may account 



240 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 



for this phase difference by an additional term in the short-circuit current 
gain [Eq. (10-28)], to give 

s exp (-jmo)/a) a ) 



<") = 



(10-33) 



1 +j(o J /w a ) 

where m has a value of about 0.2 for a diffusion transistor. The additional 
term has a magnitude of unity so the magnitude of the current generator is 



Bo- 



r bf B' 

AAA 



l-a exp (-jmu/ua ) 



Eo- 



ch 



OC 



-T* Cb'e- 



gm=^r-exp(-jmw/ Wa ) 



(a) 



-OE 




OC 



OE 



(b) 

FIG. 10-9. (a) Phase-modified hybrid-Tr circuit based on Eq. (10-33). 
(b) Hybrid-7r circuit based on a> T . 

unaltered, and only its phase is changed. For a drift or diffused-base transis- 
tor, the value of m is between 0.2 and 1 according to the value of the aiding 
field. 

A phase-modified hybrid-77- circuit based on Eq. (10-33) is shown in 
Fig. 10-9(a). In this circuit, the extra phase difference is considered to be 
contained in the a term so that wherever a appears, a exp (—jma)/a> a ) must 
be substituted. Thus, the current generator has the value of 



g m = —Qxpi-jmaj/oja) 



and r b > e is given by 



r v 



1 - a QXp(-jmco/aj a ) 

From Eq. (10-26), the alpha-cutoff frequency can be calculated. 

a exp ( —jmoj/aj a ) 
1 +jwC b > e r e 



A; = 



(10-34) 



(10-35) 



(10-36) 



SEC. 10-6 



COMPARISON OF THE HYBRID-tt CIRCUIT 



241 



Since the magnitude of the numerator is unchanged, 

1 



^b'e r e 



as before and C b > e can be chosen to conform with the measured value of w a . 
The current gain of this modified hybrid-77 circuit corresponds to the empirical 
formula, Eq. (10-33). In this way, we can retain the hybrid-77 circuit and also 
account for the difference in phase between the output and input current. 
The value of m is chosen to be appropriate to the transistor in use. In the 
analysis that follows we will consider only the diffusion transistor and put 
m = 0.2. 

For the common emitter stage, the value of f T is important. This is the 
frequency at which the magnitude of the CE current gain drops to unity. 
From Eqs. (10-11), (10-35), and (10-27), 



a exp ( —jm<x>lco a ) 
1 — a exp ( —jma>loj a ) 



1 +j 



)«[! - a Qxp(-jmoilw a )] 



a exp ( —jmco/co a ) 



1 - a QXp(-jmo)/aj a ) + j(oj/co a ) 



when \Ai\ = 1, 



and so 



ao exp (Z&2*) +y(2S) = L exp (b&2&) 



(10-37) 



(10-38) 



From the hybrid-77 circuit we know that a> T ~ a> a and, since m = 0.2, the 
exponentials can be replaced by the first two terms of a series expansion. 
Thus, 



or 



When a 
Hence, 



1 -«o+y(l +0.2a )(^) 

1, 1 - a ->0, 1 + 0.2a -> 



1.2 



,-y0.2a N 

.2, and |a - j 0.2a (co r /a> a )| a 1. 

(10-39) 



From Eqs. (10-39) and (10-32), therefore, 



OJ T zz 



1.2 ~ W 2 



(10-40) 



The more simple circuit of Fig. 10-9(b) is often used for high frequency 
and pulse analysis. The elements g m and r b . e do not include a phase term and 



242 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



the capacitance C b > e has a value which is determined from the measured value 
of co r , according to the equation 

1 



(~b'e — We i (~D ~ ^-D ~~ 



(10-41) 



It is, therefore, a simple hybrid--^ circuit based on the measured parameter 
a> T . In practice, w T will not be measured directly but will be inferred from 
measurements of gain in the CE connection at high frequencies. f T is then K 
times the frequency where the gain is K, and where K is less than 10. This 
assumes a 6 db per octave loss in gain with frequency. 

In the previous equations, we have neglected C je in comparison with 
C D . When the hybrid-77 circuit is used under small-signal, class A, amplifying 
conditions with a large emitter current, this is a justifiable assumption. If the 
emitter current is low, however, the emitter-to-base voltage is small and C je 
can no longer be ignored. In this case, as the emitter current is decreased, co T 
will fall. 



10-7 Saturation 

Saturation of a transistor can be explained by reference to Fig. 10-10. In 
part (a) of the figure, a pnp transistor is operating with a collector supply, 



A/W 




\>> 



= Vr 



(a) 




(b) 



V CE (reverse bias) 



FIG. 10-10. (a) Operating circuit for a pnp transistor, (b) Load 
line showing saturation conditions. 



SEC. 10-7 



SATURATION 



243 



V cc , and a collector load resistance, R L . The output characteristics in part (b) 
show the load line corresponding to the circuit. All currents and voltages 
shown are d-c values. 

When the transistor is operating normally, V CE is negative and the col- 
lector-to-base junction is reverse biased. As the base current is increased, the 
operating point moves up the load line from, say, Q to S. Saturation is said 
to occur at S where an increase in I B causes no significant change in I c . Now, 
using the sign convention of Chap. 8, 

h = Ic + h (10-42) 

and 

Vce = V cc + I C R L (10^3) 




Distance in base 
(a) 




Distance in base 
(b) 



FIG. 10-11. (a) Variation of hole density with distance in the base 
region of a pnp diffusion transistor operating just below satura- 
tion, (b) The same transistor in saturation. 



The value of V CE at saturation is called V cs (see Fig. 10-10), and it is 
usually a fraction of a volt. The critical value of I c when saturation occurs is 
I CSi where 

Vce ~ Vcc „ - V cc 



L* 



R, 



(10-44) 



from Eq. (10-43). Neglecting I CE0 , we can write, 

I c = h FE I B 

which applies only when I B is at, or below, the saturation value, I BS . 
base current at which the transistor goes into saturation is 



-/r.9 — 



h 



-Vc< 

hprR, 



(10-45) 
Thus the 

(10-46) 



Saturation is usually avoided in amplifiers because it produces distortion. In 
pulse operation of the transistor, it may sometimes be desirable to operate in 
saturation when the device is switched "on" since the power which has to be 
supplied to keep the transistor in this condition is low. 



244 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

Conditions in the base region under normal transistor operation (just 
below saturation) are shown in Fig. 10-1 1(a). The hole density decreases 
across the base region and falls to approximately zero at the collector-base 
junction at the right of the diagram. In Fig. 10-1 1(b), the transistor is shown 
in saturation. The saturation collector current, I cs , is proportional to the 
slope of the hole density curve at the collector-base junction as before, since 
current is still transferred across the base region. However, this current is 
limited to / cs under saturation conditions, whereas I B and I E are above the 
values necessary for saturation to occur. Thus more charge is entering the 
base region via the emitter lead than is being removed via the collector lead. 
The charge stored in the device rises to the value shown in Fig. 10-1 1(b) until 
the recombination current has increased sufficiently to give the necessary 
balance between I cs , I E , and I B . 

The horizontal dashed line drawn on Fig. 10-1 1(b) divides the stored 
charge into two parts. Q A is approximately the same as in part (a) of the 
figure if the effects of increased recombination are neglected. Q BX is the extra 
stored charge in the base region of the transistor under saturation conditions. 
When the saturated transistor is switched off, this charge must be removed 
from the base region before the collector current drops below its saturated 
value. The time for this to take place is known as the storage time and is 
discussed in Sees. 10-9 and 10-11. 



10-8 Switching the Transistor on: 
Use of the Hybrid-rc Model 

As a starting point in the discussion of the pulse operation of the transistor, 
consider the circuit of Fig. 10-1 2(a). Assume that the emitter-to-base junc- 
tion is initially biased with a reverse voltage so that no emitter current is 
flowing in the transistor. At t = 0, the base current assumes a value T bl , as 
shown in Fig. 10-1 2(b). Saturation does not occur in this circuit because 
R L = 0. Part (c) of the figure gives the hybrid-77- circuit which was derived for 
small-signal operation of the transistor at high frequences. This model will 
be used to predict how I c varies with time. Since we are using a small-signal 
model under large-signal transient conditions, we cannot expect a high 
accuracy. In this section, therefore, we shall be content to derive the form of 
the variations in output current, and so prepare for the charge control 
method of analysis given in Sec. 10-10. 

In order to switch the transistor on, the direct voltage across the base- 
emitter junction must reach a sufficiently high forward value to cause current 
to flow. This is about 0.2 volts for a germanium transistor and 0.6 volts for 
a silicon device. The capacitance in the input circuit, in the general case, has 
previously been given as 

C = C Ve + (1 + g m Rj)C h . c (10-7) 



SEC. 10-8 SWITCHING THE TRANSISTOR ON 



245 



(a) 



t/. + 



t = 
Time 
(b) 



r bb- b 
Bo W\ . o 



Eo- 



OC 



V v . 



^m V b 



\'< 



6E 



(c) 



where 



FIG. 10-12. (a) pnp transistor circuit, (b) Input current wave- 
form, (c) The simplified hybrid-7T circuit. 



^b'e — *~sje i ^D 



(10-47) 



During the time required to achieve forward conduction, however, 



gm = C D = 



because I P — 0. Therefore 



C — ^b'e + Cb'c — (->je + ^jc ~ ^H 



(10-48) 



C je will usually dominate Eq. (10-48) because the voltage across the emitter- 
to-base junction is very low (see Sec. 3-4). The resistance, r b > e , becomes in- 
finite because I E = 0, and so the input circuit reduces to r bb > in series with C je 
as in Fig. 10-13. 

The capacitance, C je , is determined by the emitter-to-base voltage and if 
V R is the reverse voltage on the emitter-to-base junction corresponding to a 
capacitance C jel , the charge initially on this capacitance is V R C jel . When the 
transistor is operating with a forward volt- 
age V b > e and its input capacitance is C je2 , 
the charge on the capacitance (of the 
opposite sign) is V h > e C m . Since the flow 
of current in the base lead, I bl , is res- 
ponsible for this change in charge, the 
delay time before collector current flows 
can be found from 

V - V b . 9 C m + V R C jel (10-49) 



Bo 



Eo- 




OE 



FIG. 10-13. Input circuit of the tran- 
sistor before initiation. 



246 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



For a small time delay, both V R and the emitter-to-base junction capacitance 
should be small. The variation in C je with voltage will be specified by the 
manufacturers for a particular transistor. When I bl is supplied from a 
voltage source with a finite source resistance, R s , the product of C je and 
Rs + f"bb' is the time constant of the circuit which represents the transistor 
while it is still in the "ofT" state. 

After emitter current starts to flow in the transistor, the simplified hybrid-^ 
circuit of Fig. 10- 12(c) can be applied. When the base current, I bx> is constant, 



h-FE h j 




FIG. 10-14. Rise o 
assumed constant). 



collective current with time (h FE and co T 



v h . e 



rises to a final value of I bl r b - e 



with a time constant of r h > e C. From 
Eqs. (10-7) and (10-47) we see that C is a function of emitter current and 
hence of time. The collector current at any time is given by 

h = gmVb'e = gmhjb'e U ~ exp ( - — ^Jj 



*■[•—(-£)] 



(10-50) 



from Eqs. (10-1 1) and (10-16). Since we are dealing with large-signal opera- 
tion, j8, which was used in the previous sections, has been replaced by h FE . In 
Eq. (10-50), the time constant, h FE la> T , varies as the current rises, since both 
C and h FE are functions of I E . This variation is considered in more detail in 
Sec. 10-10 and, in the remainder of this section, h FE and co T will be considered 
constant so that we may derive the form of the rise in output current. 

Figure 10-14 shows the collector current as a function of time, starting 
from the instant that the collector current starts to flow and ignoring the time 
delay before initiation. From the figure, t x and t 2 are the times when the 



SEC. 10- 



SWITCHING THE TRANSISTOR ON 



247 



collector current reaches 10 per cent and 90 per cent of its final value. The 
rise time, t r , is defined to be 

(10-51) 



From Eq. (10-50), we see that the final collector current is h FE I bl , and so 



and 



giving 



and 



Hence, 



--OS?) 

/ - f 2 cu r \ 
\ h FE ) 



0. 



0.9 - 1 - exp 



h 



^log e 0.9 = 0.1^ E 



a> T 



to T 



t 2 = — log. 0.1 = 2.3^ 



U 



to. - U 



2.2 



cd t 



(10-52) 



(10-53) 



(10-54) 



(10-55) 



(10-56) 



It is seen from Eq. (10-56) that, for a given large-signal gain, a high value of 
f T produces a low rise time. 

There are three major omissions in this very simple analysis. It does not 
account for variations in a> T and h FE with voltage and current changes; it 
does not consider the effects of a load in the collector circuit; it is not directly 
applicable when the transistor goes into saturation. The waveforms which 
are obtained experimentally when switching the transistor on and off when 
saturation is reached are given in the next section: a more complete analysis 
which takes the omissions into account is presented in Sec. 10-10. 



10-9 Definition of Switching Times 

for the Transistor: Saturation Case 

A common method of pulsing the transistor is to control the base current 
in a CE circuit so that the transistor can be turned on or off. Figure 10-1 5(a) 
shows a schematic circuit that will produce a suitable base current waveform. 
When the switch is in position A, a constant current, I bl , flows in the base 
lead, and when the switch is in position B the base-emitter junction is reverse 
biased. If l bl is high enough to saturate the transistor, the collector current 



waveform of Fig. 
on the figure : 



10-14(b) is obtained. There are four time intervals indicated 



248 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



QB 

Vr 



AA/V 



■^rVr 



(a) 




J 



S3 

en C 



Time — * 
(b) 

FIG. 10-15. (a) Switching circuit for a pnp transistor, (b) Input 
and output current waveforms when saturation achieved. 

t d , the delay time, is the time which elapses between the onset of base current 
and the rise of the collector current to 10 per cent of its final value. 

t r , the rise time, is the time it takes the collector current to increase from 
10 per cent to 90 per cent of its final value. 

t s , the storage time, is the time required to remove excess charge (Q B x m 
Fig. 10-1 1) from the base region before the transistor comes out of satura- 
tion. It is defined as the time which it takes the collector current to fall to 
90 per cent of its saturation value after the base current is switched off. 

//, the fall time, is the time for the collector current to drop from 90 per cent 
to 10 per cent of its saturation value. 

If the transistor is not allowed to saturate, the storage time is very small. 
Since the total time required to switch the transistor off is / s + t f , it would 
appear that the transistor should never be operated in saturation for fast 
switching off. However, many applications arise where it is desirable to have 



SEC. 10-9 



DEFINITION OF SWITCHING TIMES FOR THE TRANSISTOR 



249 



the "on" condition of the transistor in saturation since this usually results in 
more simple circuitry and less power loss in the device. Overdriving the base 
circuit, which is necessary for saturation, also has the advantage that it re- 
duces the rise and fall times as shown in Sec. 10-1 1. 

10-10 Charge Control of the Transistor 

In this section, the elementary concept of base charge control of the transistor 
is developed. This method of analysis leads to a greater understanding of the 
operation of the transistor and is of great value in predicting switching times. 




FIG. 10-16. Minority and majority charge in a diffusion pnp 
transistor under three conditions of operation. 

Figure 10-16 gives the minority and majority charge distributions in the 
base region of a. pnp diffusion transistor under three conditions of operation. 
When both the collector-base and emitter-base junctions are reverse biased, 
part (a) of the figure shows that the minority carrier density is small over the 
whole width of the base and that it falls off close to the two reverse-biased 
junctions. Under small-current conditions the majority charge density is 
assumed to be unchanged across the base region as shown in Fig. 10-16(a). 
Assuming that no acceptor atoms are present in the ft-type base material, we 
can write 

"nO = PnO + N d x N d (10-57) 

since n n0 » p n0 in practice. The majority carrier density in the base is 
constant at approximately the value N d . 

Figure 10- 16(b) shows the base charge when the transistor is operating 
with a high emitter current. This current is injected into the base from 



250 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 



the emitter region and substantially increases the minority charge density 
close to the emitter-base junction. The current is carried across the base 
entirely by diffusion, and so the minority charge drops linearly with distance, 
falling to a value of almost zero at the collector-base junction. Because of 
charge neutrality, the majority charge density has the profile shown in Fig. 10- 
16(b), that is, it is everywhere increased above its thermal equilibrium value 
by an amount equal to the increase in the minority charge density. (See 
Sec. 9-7 for a brief discussion of charge neutrality under these conditions.) 
We can write, 



and so from Eq. (10-57), 



n n = Pn + N d 



"n - n n0 = Pn - PnO 



(10-58) 



(10-59) 



The final diagram, Fig. 10- 16(c), shows the base charge density when the 
transistor is in saturation. In this case, the minority charge at the collector 
end of the base region is not zero, and there is a large amount of minority 
charge stored in the base region. Equation (10-59) is still obeyed, giving the 
charge profile shown. 

When the transistor is switched from the "off" to the "on" state, the 
charge contained in the base region must be changed from the value shown 



h:' 



Ooff 



C L 

He- 

-WW 

Rl 



FIG. 10-17. Circuit for switching the transistor on. 

in Fig. 10-1 6(a) to the value shown in part (b). The transient operation of the 
transistor is studied here by analyzing the way in which the charge in the base 
region changes from one state to another. In our analysis, we shall neglect 
the effects of transit time and concentrate on the times required to build up 
or remove the base charges Q A and Q BX . 

When &pnp transistor in the CE connection is switched "on," the current 
in the base lead results in a flow of electrons into the (n-type) base region. The 
majority charge density is increased by the current flow in the base lead since 
no appreciable electron flow occurs at the two junctions. To obtain the 
collector current, I c , appropriate to the conditions shown in Fig. 10-1 6(b), 
sufficient majority charge (electrons) must have entered the base region (via 
the base lead) to build up the required charge profile. This is the dominant 



SEC. 10-10 



CHARGE CONTROL OF THE TRANSISTOR 



251 




Distance in base — ► 

FIG. 10-18. Active charge in the base 
region for a diffusion transistor. 



factor in determining the rise time. As 
the majority charge in the base region 
increases, the minority charge rises so 
that charge neutrality is still obeyed. 
The increase in minority charge density 
results in the flow of emitter current 
(i.e., holes across the emitter-to-base 
junction) and so there is little delay 
between the rise in majority charge 
density and the corresponding rise in 
the minority charge density. The col- 
lector current rises in unison with the 
increase in minority and majority charge 
densities until the final collector current 
is reached. 

The rise time for the collector cur- 
rent can be determined by computing 

the amount of charge which is supplied to increase the active majority charge, 
Q A . All of the charge supplied by the base current is not used for this purpose, 
however, because the junction capacitances and the recombination mechan- 
ism require additional charge. For the circuit of Fig. 10-17, the base current 
supplies five components of charge during the rise time of the collector 
current. These five components are listed and discussed below. 

1. Active charge in the base region, Q A . 

2. Charge on the emitter-base junction capacitance necessary to change 
the junction potential to its operating value. 

3. Charge on collector-base capacitance to correspond with the change 
in V cb . 

4. Charge to replenish that part of the active charge lost by recombina- 
tion. 

5. Charge required to change the voltage across the stray capacitance of 
the load (see Fig. 10-17). 

1. The active charge in the base of a diffusion transistor is shown in 
Fig. 10-18. Neglecting recombination, the emitter current for a base of area 
A is 



h = eAD p 



Pe 

w 



(10-60) 



where p e is the hole density at the emitter end of the base region. The active 
charge from Fig. 10-18 is 

ep e AW W 2 _ 



Qa = 



2D, 



(10-61) 



252 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

and so the amount of active charge in the base region is proportional to the 
flow of collector current. Assuming W 2 /2D P is constant (i.e., neglecting base 
modulation), 

dQA= S~ P dic (io_62) 

From Eqs. (4-45) and (5-15), we can write 

W 2 
C D r e = jo- (10-63) 

Hence, 

dQ A = C D r e dl c (10-64) 

This value will be used in the next paragraph. 

2. As soon as the collector current starts to flow, the base-to-emitter 
voltage has a small magnitude which will increase throughout the rise time. 
During this period, charge is being supplied to the base-to-emitter capacitance, 
C je . Neglecting recombination in the base, an increase in emitter current, 
dl e , causes a change in emitter-to-base voltage of r e dl e , and so the increased 
charge on the capacitance is 

dQje — C je r e dl e & C je r € dl c (10-65) 

The sum of the charges in paragraphs 1 and 2 gives 

dQ A + dQ je = (C D + C je )r e dl c 

= — dl c (10-66) 

from Eq. (10-41). 

3. A change in collector current of dl c causes a voltage change of R L dl c 
at the collector terminal. The change in charge on the collector-to-base 
junction capacitance, C jc , is therefore 

dQ JC = C jc R L dl c (10-67) 

assuming V be « V ce . 

4. When a transistor is operated with a constant current / c , the base 
current is given by IJh FE where h FE is the large-signal value of the current 
gain. Thus a base current of I c /h FE must be supplied to the base lead to re- 
plenish the active charge lost by recombination. In a time dt, the charge 
supplied to the base for this purpose is dQ R , where 

dQ R = ^-dt (10-68) 

n FE 

5. If the load resistance is shunted by the stray capacitance C L as shown in 
Fig. 10-17, a change in collector voltage of R L I C requires a change in charge 



SEC. 10-10 CHARGE CONTROL OF THE TRANSISTOR 253 

on the capacitance of C L R L dl c . This charge is supplied from the collector-to- 
emitter circuit. Therefore, the charge which has to be supplied to the base 
region to change the charge on the load capacitance by C L R L dl c is 

dQ c = ^dI c (10-69) 

All five components of charge are supplied by the base current, therefore 
I b dt = dQ A + dQ je + dQ ic + dQ R + dQ c (10-70) 

Substituting from Eq. (10-66) through (10-69), 



h = 



t + (i +c > A+£ &)§ (10 - 71) 



In this equation, h FE , co T , and C jc are functions of I c . If we assume that at 
t — 0, I b rises instantaneously to a value I bl , then over a small range of values 
I c such that h FE , to T , and C jc can be considered constant, Eq. (10-71) can be 
solved to give 

/.-A„/ tx [l-«p(jg)] (10-72) 



where 

+ C ]C R L + ^P- L (10-73) 



1 . ^ . . C L R L 



CO 



Equation (10-72) can be compared with the solution for the simple hybrid-?! 
circuit given in Eq. (10-50). 

The time constant, r, will vary as the collector current rises. In general, 
t is not a strong function of I c and, as a first order approximation, we may 
write, 

r RE = f (10-74) 

where f is the mean value of r. Experimentally, r RE can be measured for a 
particular transistor by finding the rise time under given conditions. (See 
Eig. 10-20 in Sec. 10-11 for a curve of r RE measured under saturation con- 
ditions and for comments on the variation in t re with I cs .) From Eqs. (10-50) 
and (10-56), 

t r = 2.2h FE r RE (10-75) 

where h FE is the mean value of the current gain over the current range 
concerned. 

It is possible to calculate the variation in the individual terms of r RE as 
I c or V cc is varied. This can be done if the curves of oj t , h FE , and C jc are 
known. The method can then be used to provide a specification of the transis- 
tor under pulsed conditions of operation. 

The fall time of the transistor is almost the same as the rise time since the 
transistor is operating over the same part of its characteristics. The delay 



254 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

time is composed of two time periods. The first of these is the time elapsing 
between the switching on of the base current and when the base-to-emitter 
voltage just reaches the forward direction. This time was calculated in 
Sec. 10-8. To this must be added the time taken for the collector current to 
rise to 10 per cent of its final value, which can be calculated from a knowledge 

Of r RE . 

From this brief account, it can be seen that this is far from being an exact 
method of analysis. Transistors have so many interrelated parameters, how- 
ever, that approximations must always be made and errors of up to 20 per 
cent are permissible. Further details of the transient response characteriza- 
tion of transistors can be found in the two papers by Ekiss and Simmons 
cited in the bibliography at the end of the chapter. 

10-1 1 Switching Times for a Saturated Transistor 

The transistor shown in Fig. 10-17 will saturate if 

h FE I bl > Ics * -^ (10-76) 

In this case, the rise time of the collector current is the time elapsing between 
the attainment of 10 per cent and 90 per cent of the saturation current I cs . 
From Eq. (10-72), and substituting r RE for the average time constant, we have 

0.1/« = W*. [' - exp (j^~j\ (10-77) 

0.9/ cs = knit, [l - exp (j^)] (10-78) 



and 

Thus, 

t r = U 



For the case when the transistor is driven well into saturation so that h FE I H > 
5 l cs , Eq- (10-79) can be written in an approximate form by using a series 
expansion and retaining only the significant terms. Then, 

r r = 0.8r B£ ^ (10-80) 

Figure 10-19 shows the rise in collector current when the transistor is well 
in saturation. The collector current follows the dotted line of the initial slope 
until I cs is reached. Increasing the base drive (I bl ) for a given saturation 
current decreases the rise time. 

The variation of r RE with I cs is shown in Fig. 10-20 for a medium fre- 
quency, alloy-junction transistor. The curve was obtained by measuring the 



SEC. 10-11 SWITCHING TIMES FOR A SATURATED TRANSISTOR 



255 



Inital slope = h EE t re 




Time — *■ 

FIG. 10-19. Rise of collector current when transistor is well in 
saturation. Rise time, t r , is time between 10 per cent and 90 per 
cent I cs . 

rise time for the saturated case as a function of I cs - The ratio I C s/hi was 
constant throughout the measurements. At small values of I cs , r RE is high 
because the term C jc R L dominates Eq. (10-73). As I cs increases, r RE dimi- 
nishes because w T increases and the other terms decrease. When I cs is large, 
r RE slightly increases again. At the vicinity of the broad minimum ofr REi the 
term C jc R L is typically about 10 per cent of the total value. R L CJh FE is 
assumed to be negligible in this case. 



140 



100 - 



80 



60 



20 - 



8 volts 



20 



*a 



FIG. 10-20. Variation in r RE with I cs for a low frequency tran- 
sistor. [From J. A. Ekiss and CD. Simmons, "Junction Transistor 
Transient Response Characterization," Solid State Journal, 2, 1 
(January 1961), p. 20.] 



256 HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR CHAP. 10 

When a saturated transistor is switched off, the excess charge Q BX (shown 
in Fig. 10-11) must be removed from the base region before the collector 
current will fall again. The base current which will just saturate the transistor 
is 

Ibs = jf < 10 - 81 > 

n FE 

Therefore, the transistor may be brought out of saturation (though it will not 
be switched "off") if the base current is dropped below I BS . If the base current 
is dropped to zero, the transistor will eventually return to the "off" state. 
However, there is a third possibility. This is that the base current may be 
reversed for a short time to remove the charge in the base region more quickly 
and so reduce the storage time. All three possibilities are accounted for in the 
following equations. 

Let Q be the total charge in the base region at any time, r, where Q includes 
Q A and Q BX of Fig. 10-1 1. When a current I b is flowing in the base lead, we 
may write 

where t s is the lifetime of the charge carriers in the base region, and we have 
neglected the charges on the capacitances. When a current I bl is flowing and 
the transistor is saturated, dQjdt — 0, and 

V. = Q = Qa + Qbx (10-83) 

We shall now examine the conditions during the storage time. Let base 
current I b2 flow in the same direction as I bl . Equation (10-82) can now be 
written as 

$ + ?-'- <«"« 

and a solution is of the form 

Q = h 2 r s + Cexp(^) (10-85) 

where C is a constant to be determined. Assume that I b2 starts to flow at 
t = and that up to this time current I bl was flowing. Then by comparing 
Eqs. (10-83) and (10-85), at t = 0, 

h{r s = Q = h 2 T s + C 
giving, 

C = (I bl - I b2 )r s (10-86) 

Equation (10-85) now becomes 

Q = V. + (I bl - I b2 )r s exp (^) (10-87) 



SEC. 10-11 



SWITCHING TIMES FOR A SATURATED TRANSISTOR 



257 



The transistor just comes out of saturation when Q BX = and Q = Q A . 
From Eqs. (10-81) and (10-82), 

Qa = IbS t s 



(10-88) 



Hence, 



Ws = h 2 r s + (I bl - h z )r s exp (-— s j 



or 



t s = r s log, 



Vbs - I J 



(10-89) 



where t s is the storage time. 

Equation (10-89) is only approximately obeyed in practice. The lifetime, 
r s , is the value applying to the saturated condition of operation of the 
transistor. It is usually measured by finding the storage time, / s , when 



^-Ji (negative value) 




Time 



FIG. 10-21. Storage time when I b2 = and when I b2 is negative. 



I bl , I b2 , and I BS are known. r s is usually below a microsecond for high 
frequency transistors and varies somewhat with I BS . 

The effect of changing the switching-off base current, I b2 , is illustrated in 
Fig. 10-21. The full line shows the condition I b2 — 0, and the dotted line 
shows that the storage time is reduced when I b2 has a negative value as can be 
seen from Eq. (10-89). 

The circuit of Fig. 10-22 illustrates the use of a "speed-up" capacitor C. 
When an input step voltage is applied, the initial base current is high, and the 
rise time of the circuit is reduced. This type of circuit can also be used to 



258 



HIGH FREQUENCY AND PULSE OPERATION OF THE TRANSISTOR 



CHAP. 10 




Time 




= V C 



FIG. 10-22. Use of "speed-up" capacitor to reduce rise time. 

reduce storage time by effectively removing the stored base charge when the 
transistor is switched off. 



BIBLIOGRAPHY 

Hybrid-7T circuit 

Cote, Alfred J., Jr. and J. Barry Oakes, Linear Vacuum-tube and Transistor 
Circuits, New York: McGraw-Hill Book Company, Inc., 1961 

Joyce, Maurice V., and Kenneth K. Clarke, Transistor Circuit Analysis, 
Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1961 

Pettit, Joseph Mayo, and Malcolm Myers McWhorter, Electronic Amplifier 
Circuits, Theory and Design, New York: McGraw-Hill Book Company, 
Inc., 1961 

Pulse operation 

DeWitt, David and Arthur L. Rossoff, Transistor Electronics, New York: 
McGraw-Hill Book Company, Inc., 1957 

Ekiss, J. A., and C. D. Simmons, "Junction Transistor Transient Response 
Characterization," Parts I and II, Solid State Journal, 2, 1 (January 1961), 
17-24; 2, 2 (February 1961), 24-29 



PROBLEMS 

1 0-1 Calculate the Miller capacitance C for the transistor of Prob. 9-4. 
What is the gain-bandwidth product for this transistor? 

10-2 Find the collector load resistance for the hybrid-7r circuit of Fig. 10-1 at 
which oj 3db = oj /2. 

ans. 3900 ohms 

10-3 Fig. 10-1 shows the value of the components of the hybrid-77- circuit 
when I E = 1 ma and V CE = — 5 volts. What are the component values 
when I E = 10 ma and V CE = - 10 volts? What effect does this have 
on a>„? 



PROBLEMS 259 

10-4 A manufacturer makes a diffused base transistor which can be ade- 
quately represented by a modified hybrid-77 circuit where 



1 — a exp (— j0.5oj/co a ) 



«o 
g m = - exp 



(=5*0 



Assuming that the frequency of operation is considerably smaller than 
co Ti show that there is a frequency range where the gain of the transistor 
drops at the rate of 6 db per octave and compute the phase shift through 
the device as a function of frequency. 

10-5 How would you expect the value of w T to vary with V CE for an alloy 
junction transistor? Give a physical reason for all the factors which 
are discussed and briefly define all new terms that are introduced. 

10-6 A CE transistor stage can be represented by a hybrid-77 circuit. Assign 
typical values to all the elements and then consider how this equivalent 
circuit can be used to determine gain and frequency response of a 
multistage amplifier. Calculate these two quantities using typical values 
of resistances and capacitances, (a-c circuit only required.) 

10-7 Explain how the methods of charge control can be applied to the 
understanding and specification of transistors for pulse operation. 

10-8 Using the information given in Sees. 10-10 and 10-11, calculate the 
rise time for a transistor having the r RE characteristic shown in Fig. 
10-20, under the following conditions: (a) I bl = 10 ma, V cc = — 8 volts, 
R L = 320 ohms, (b) I bl = 10 ma, V cc = - 8 volts, R L = 4000 ohms. If 
h FE is considered to be a constant of value 50, what is the maximum 
capacitance that can be added across the load to increase the rise time 
by 10 per cent in both cases? 

10-9 The transistor of Prob. 10-7 is operating in saturation with V cc = — 8 
volts, R L = 320 ohms and I bl = 10 ma. When the base current is 
suddenly reduced to zero, the storage time is 200 nsecs. What is the 
value and direction of the base current necessary to reduce this time to 
100 nsecs? 



11 



The The tunnel diode is an important solid-state device 
that was discovered by Esaki in 1958. It consists of 
Tunnel Diode a heavily doped /?-« junction in which the transition 
from p- to fl-type material occurs over a distance of 
less than 10 -6 cms (100 A). As is shown later in 
this chapter, this device can be used as an amplifier, 
an oscillator or a switch. The tunnel diode has 
many applications in fast electronic circuitry since 
it is capable of operating at a frequency in excess of 
10 10 cycles per second and of switching in a time 
less than 10 ~ 9 seconds. Among the advantages of 



SEC. Il-I 



QUANTUM MECHANICAL TUNNELING 



261 



the tunnel diode are its small size, its low voltage and power requirements, a 
very wide temperature range of operation, low cost and ease of fabrication, 
and its extremely high speed of operation. The principal disadvantage of the 
device is that it has only two terminals, and so its equivalent circuit over part 
of its characteristic is a one port active network. Thus, the input and output 
circuits cannot be isolated and precautions must be taken to avoid unwanted 
high frequency oscillations. 

The tunnel diode gets its name from the "tunnel" effect which is a pheno- 
menon that can only be explained by quantum mechanics. Accordingly, the 
first section of this chapter gives a brief quantum mechanical picture of an 
electron "tunneling" through a potential barrier. 



Il-I Quantum Mechanical Tunneling 

In classical physics, a particle with total energy E cannot pass into a region 
where the potential energy, U, has a higher value. Thus, a ball of mass m with 
kinetic energy \mv 2 can travel up a hill to a height h given by \mv 2 = mgh, 



Potential 
barrier 



Incident 
electrons 



Reflected 
electrons 



~~\ 



U 



Transmitted 
electrons 



FIG. I 



Diagram of electrons incident on a potential barrier. 



where g is the acceleration due to gravity, but cannot exceed that height. How- 
ever, many phenomena in nuclear and atomic physics can only be explained 
by assuming penetration of such a high barrier by a subatomic particle. Using 
quantum mechanics, it can be shown that there is a small but finite proba- 
bility of a particle passing through a higher energy barrier. This is known as 
the tunnel effect. 

Figure 1 1-1 shows a beam of electrons of energy E incident on a barrier 
of potential energy U. As shown in the diagram, E < U. By referring back 
to Sec. 1-8, we see that the behavior of the incident electrons can be de- 
scribed by the wave function, ifj. The wave function in the x direction is the 
solution of the Schrodinger wave equation [Eq. (1-15)]. This equation is 
repeated here for convenience. 



d 2 ijj %ir 2 m fr . . 



(n-i) 



262 



THE TUNNEL DIODE 



CHAP. II 



Particles traveling in the positive x direction in free space may be rep- 
resented by waves having the alternative forms : 

ifj = A exp [j(wt - ex)] (11-2) 

or i/j = A exp [-j(<ot - ex)] (11-3) 

We will use the second form and suppress the time-dependent term. Electrons 
traveling in the positive x direction will, therefore, be represented by 

ijj = A exp(jcx) (11-4) 

Electrons moving in the negative x direction can be represented by 

+ = Bexp(-jcx) (11-5) 

On the left-hand side of the potential barrier of Fig. 11-1, we can postulate 
that A 1 is the amplitude of the incident wave and that B 1 is the amplitude of 



Region 1 Region 2 

Incident + reflected wave 



Region 3 




FIG. 11-2. Diagrammatical representation of tunneling. 

the wave reflected from the potential barrier. We can, therefore, write the 
wave function in the form 



where 



0! = A x exp (jc ± x) + B 1 exp (-jc ± x) 
%7T 2 mE 



r 2 — 



(11-6) 



(11-7) 



since U = in this region. The energy of the incident electrons (E) is 
positive, and so c 1 is real and the wave motion is as represented in region 
1 (x < 0) in Fig. 11-2. 

The potential barrier exists between x =0 and x =a and is designated as 
region 2. Here the value of E — U is negative, and so the exponents in the 
solution of the wave equation are real. Writing 



4 = *p(U-E) 

where c 2 is defined in such a way as to be real, we have 
ifj 2 = A 2 exp (c 2 x) + B 2 exp ( — c 2 x) 



(11-8) 



(11-9) 



SEC. Il-I QUANTUM MECHANICAL TUNNELING 263 

We note that in many practical applications (such as the tunnel diode), 
c 2 a » 1. Hence, A 2 « B 2 since ^->0 as a-> oo. Note that the electrons 
cannot be detected in this region since the wave function is not oscillatory. 

In region 3, x > a, U — and there will be a transmitted wave only in 
the positive direction of x. Therefore, the solution of the Schrodinger wave 
equation is 

03 = A 3 Qxp(j Cl x) (11-10) 

where c x = %-n 2 mE\h as before. 

From Eq. (11-10), we see that A 3 is the amplitude of the wave function 
representing the transmitted beam of electrons. Provided A 3 ^ 0, there will 
be some probability of electrons "tunneling'' through the potential barrier 
since \A 3 \ 2 is proportional to the number of transmitted electrons (see 
Sec. 1-8). 

The relationship of the amplitudes A 1 , B l9 A 2 , B 2 , and A 3 can be found by 
applying the conditions that and dift/dx are continuous at the boundaries 
between the regions. This is a necessary mathematical condition relating to 
Eq. (11-1) since ip, E, and U are finite, i/j must be finite since it is related to 
the probability of finding particles at a particular location; E and U are finite 
in all practical cases. Applying these conditions at x = and x — a and 
eliminating B l9 A 2 , and B 2 from the resulting equations allows us to determine 
the ratio of A 3 to A x . We will not carry out this procedure here, but it can be 
shown that the ratio of the intensities of the transmitted and reflected waves, 
when A 2 « B 2 , gives a transmission coefficient T, where 

T \M 2 _ 4exp(-2c 2 a) m-in 

■ m,I 2 i + i(c 2 /c x - cjc 2 y U1 11; 

Thus, T is the fraction of the electrons in the incident beam "tunneling" 
through the barrier. 

Equation (11-11) shows that there is a finite but small probability that 
particles will cross a potential barrier even though their initial energy is 
smaller than the barrier height. This probability decreases exponentially with 
the width of the barrier (a) and also decreases as the barrier height increases 
[Eq. (11-8)]. Figure 11-2 shows the wave functions diagrammatically in the 
three regions. It should be noted that both the incident and the transmitted 
electrons have the same energy E. Tunneling will only take place when 
electrons possess the same energy on each side of the barrier. 



11-2 Energy Bands of the Tunnel Diode 

The energy bands of a tunnel diode can be obtained by reference to those of a 
conventional p-n junction. If the acceptor and donor densities on the two 
sides of a junction are less than about 10 18 cm -3 , Fig. ll-3(a) applies. On 



264 



THE TUNNEL DIODE 



CHAP. II 



each side, the Fermi level (defined as that energy where the probability of 
occupation is one half) lies in the forbidden gap (see Sees. 2-4 and 2-5). 

Figure 1 1— 3(b) shows the energy bands for the tunnel diode. On the 
/7-type side, the donor density is greater than 10 19 cm -3 , and thus a large 
number of electrons is present in the conduction band. The Fermi level for 
this case lies within the conduction band. Similarly, the acceptor atoms on 
the p-type side are so numerous that there are many holes in the valence band 
and the Fermi level is within that band. 

At room temperature, the presence of the Fermi level within the con- 
duction band of the n-type material means that most of the electron states 
at the bottom of the band will be occupied. The distribution of filled states 
can be calculated from Fermi-Dirac statistics (see Chap. 2). However, the 
Boltzmann factor that was used in Chap. 2 as an approximation to the 



p-type 



"-type 



p-type 



n-type 




Fermi 
level 



Valence 
band 



Partially unfilled 

at room 

temperature 

Fermi Fermi z/7 

level level //A 




Conduction 

band y- Fermi 

Z2ZZZ ^ level 

Partially filled 

at room 

temperature 



Distance 
(a) 



Distance 
(b) 



FIG. 11-3. (a) Energy bands for a conventional p-n junction (zero 
applied voltage and room temperature), (b) Energy bands for the 
tunnel diode where both impurity densities are greater than 10 19 
cm -3 (zero applied voltage and room temperature). 



Fermi-Dirac distribution is no longer applicable since there are now states 
close to the Fermi level. Thus, the Fermi factor as given in Eq. (2-10) must 
be used. This is a condition known as degeneracy. By this definition, it is 
seen that both the n- and/?-type material used in a tunnel diode are degenerate. 
Whatever the temperature, the Fermi levels on the two sides of the junc- 
tion will line up. For the tunnel diode, therefore, tunneling will take place 
where the two bands overlap in their energy ranges. This has been indicated 
on Fig. 1 1-4. Here, it is possible for electrons to tunnel through the potential 



SEC. 11-2 



ENERGY BANDS OF THE TUNNEL DIODE 



265 



barrier separating the conduction band of the tf-type material from the valence 
band of the/?-type material. The tunneling current will be proportional to the 
product of the probability of occupancy on the side the electrons leave, the 
probability of nonoccupancy on the side they reappear, and the transition 
probability. When there is no applied voltage, the net tunneling current is 
zero since the probability of occupation on the two sides is the same at the 
same energy level and the probability of tunneling in either direction is the 
same. 

A calculation of the tunnel current flowing in either direction can be made 
by the application of quantum mechanics. The analysis must take into 
account the variation in the height of the potential barrier with distance, since 



p-type 

t 

& 

a> 

C 

H 

Potential ^ 

barrier""^ I \, 
I \ 
Fermi I \ 

level -*■ 



"-type 



p-type 



n-type 





zzzr™ 



Fermi 




V777/7/7 -^™ 
level 



Distance -»>■ 

FIG. 11-4. Energy bands of the tunnel 
diode showing tunneling at room temper- 
ature and zero applied voltage. 



Distance — ► 

FIG. 11-5. Energy bands of the tunnel 
diode at room temperature and reverse 
bias condition. 



the barrier is formed by the lower edge of the conduction band across the 
junction (shown dotted). For tunneling to be appreciable, the width of 
the junction must be small. This condition can be achieved in an abrupt 
p-n junction which is doped to degeneracy on both sides of the junction. 

M-3 Forward and Reverse Characteristics 



When a voltage, V, is applied to the junction, the Fermi levels on the two sides 
are spaced eV apart on the energy band diagram. Figure 1 1-5 shows the case 
where V is negative. Here, the reverse voltage has depressed the Fermi level 
on the tf-type side by an energy eV relative to that of the /?-type Fermi level. 



266 



THE TUNNEL DIODE 



CHAP. II 



The number of electrons which can tunnel to the right is much increased from 
the case where V = 0. This is because electrons from the numerous occupied 
states close to the Fermi level on the left-hand side, and above the Fermi level 
of the right-hand side, can tunnel into the relatively unoccupied states at the 
same level on the right. Thus, the reverse current of the tunnel diode is high, 
as shown below in Fig. 1 1-6. 




Fermi 
level 



Fermi 
level 



Distance — +~ 
(a) 



Distance- 
(b) 



Electron 
injection 
current 



Conduction band 
n-type 

w///'i"' 




Forward 
current i" 




Reverse 
voltage 

Reverse tunneling 
(Fig. 11-5) 



Forward 
voltage V 



Reverse 
current / 

(d) 
FIG. 11-6. Detail of energy bands and tunnel diode characteristic. 

When the junction is biased in the forward direction with a voltage of a 
few millivolts, the tunneling current of electrons from n- to /?-type is greatly 
increased. This is shown in Fig. 1 l-6(a). In this case, there are many filled 
states at the bottom of the conduction band of the w-type material in line 
with unfilled states in the valence band of the/7-type. The number of electrons 
tunneling in the opposite direction is much smaller. 

In Fig. 1 l-6(b), a higher forward voltage has been applied. Consequently, 
the amount by which the two bands overlap in energy has been reduced. 



SEC. I 1-3 FORWARD AND REVERSE CHARACTERISTICS 267 

There is a corresponding reduction in the tunnel current since some of the 
filled energy states just below the Fermi level in the «-type conduction band 
are now above the top of the p-typt valence band. Electrons in these states 
cannot tunnel since there are no available states on the opposite side to receive 
them. Therefore, a further increase in forward voltage results in a decrease in 
tunneling current in the forward direction. As the voltage is increased, 
tunneling ceases and eventually the conventional injection current of the p-n 
junction predominates as shown in Fig. 1 l-6(c). 

The characteristic curve given in Fig. 1 1— 6(d) shows the values I p and V p 
for the peak tunneling current and voltage, and also I v and V v for the current 
and voltage at the "valley" of the curve. The voltage swing, V s , is also de- 
fined on this characteristic. It is the voltage between the two points on the 
curve where the current is equal to I p . The value of V s is a characteristic of 
the material as shown in Table 1 1-1. 



TABLE 1 1 -I Properties of tunnel diodes. 

Material Peak-to-valley V s Energy gap 

current ratio (volts) E g in ev 

I P /Iv 

Ge 10-30 0.45 0.72 

Si 3-5 0.7 1.1 

GaAs 30-60 1.0-1.2 -1.4 



The reduction in tunnel current as the voltage is increased results in a 
negative resistance region. In this region, power is transferred from the supply 
to the external circuit. It is this property of the tunnel diode which makes it 
useful as an amplifier, an oscillator or a switch (see Sec. 11-4). 

Tunnel diodes can be made of the semiconducting elements, germanium 
or silicon. However, gallium arsenide (GaAs) and other group 3-5 com- 
pounds are also used. These are materials where the group 3 and 5 atoms 
form crystalline compounds having properties similar to the group 4 semi- 
conductors. All types of tunnel diode exhibit basically similar characteristics; 
for example, the value of the peak current increases with the junction area. 
Peak currents above one hundred amperes have been obtained in germanium 
units of large area. The voltage range of the tunnel diode depends on the 
particular semiconductor used, and little variation is possible for any one 
type of material. Values of the peak-to-valley current ratio, the voltage swing, 
and the energy band gap for three types of tunnel diode are tabulated in 
Table 11-1. It is seen that the negative resistance region occurs below one 
volt for all three types. 

Since tunnel diodes are fabricated with very heavily doped semiconductors, 
it is not absolutely necessary to use very pure material in their manufacture. 
In addition, in many cases, surface conditions are much less critical than in 



268 



THE TUNNEL DIODE 



CHAP. II 



transistor fabrication. Thus, many types of tunnel diodes are easier and 
cheaper to manufacture than transistors. However, special types of tunnel 
diodes with tight parameter tolerances and special performance properties 
can be as expensive as any other semiconductor device. 

The negative resistance characteristic of the tunnel diode may be ob- 
served over quite a wide temperature range. Both high and low temperature 
operation are possible since the device is heavily doped on both sides of the 
junction and no appreciable (temperature-sensitive) minority carrier injection 
occurs during tunneling. For germanium units, a temperature range of 4.2° K 
to 250° C has been observed; for silicon, the upper temperature limit is ex- 
tended to about 350° C. The peak-to-valley current ratio is temperature 
sensitive and varies widely among different units. 

11-4 The Tunnel Diode as a Circuit Element 

There are two important factors which govern the use of the tunnel diode in 
electronic circuits. In the first place, the limitations in switching time and 
frequency of operation are those imposed by the equivalent and external 



External 
circuit 



(a) 



AA/V 



L s 






-G 



(b) 




(c) 



Zf 



Rt 

AyW 



^ffuTP r 



FIG. 11-7. (a) Equivalent circuit of tunnel diode, (b) Tunnel 
diode and external circuit, (c) Equivalent circuit of the combina- 
tion. 



circuit of the diode, and the inherent limit set by the tunneling mechanism is 
never approached. Nevertheless, switching times of less than 10~ 10 seconds, 
and oscillation frequencies above 10 10 cycles per second are possible in units 



SEC. 11-4 THE TUNNEL DIODE AS A CIRCUIT ELEMENT 269 

having high peak current to capacity ratios, I p /C (values as high as 200ma/pf 
have been achieved in germanium. Secondly, the tunnel diode is a one port 
(two terminal) network and this raises difficulties in operation since the input 
and output circuits cannot be separated. 

When the tunnel diode is operated in the negative resistance part of its 
characteristic, the equivalent circuit for small-signal operation is as shown in 
Fig. 11 -7(a). R s is the ohmic resistance of the semiconducting material, and 
the construction of the device determines the small series inductance, L s . C is 
the junction capacitance due to the depleted charge regions on the two sides 
of the p-n junction. The negative resistance has been represented by a con- 
ductance — G. Typical values for these quantities are: R s = 2 ohms, L s = 
10~ 2 microhenries, C = 10 picofarads, — G = —0.01 mhos. 

Assume that the tunnel diode is operating with an external circuit having 
a series resistance R 1 and a series inductance L x as given in Fig. 1 1— 7(b). In 
part (c) of this figure the equivalent circuit of the combination is shown, 
where L t = L\ + L s and R t — R 1 -+- R s - Note that this final circuit is 
similar to the equivalent circuit of the tunnel diode alone, where R s is re- 
placed by R t and L s is replaced by L t . Thus, the analysis of the complete 
circuit applies equally to the tunnel diode under short-circuit a-c conditions 
when R s and L s are substituted for R t and L t . 

The impedance, Z h at an angular frequency co, is given by 

Z t = R t + jcuL t + ! . (11-12) 

(-G) + jojC 

or 

Z > = R > + G^%> + i" L > - (FT&c*) (IM3) 

Note that both the real (resistive) and imaginary (reactive) parts of the ex- 
pression are frequency dependent. 

There are two important frequencies for this circuit. w R is the resistance 
cut-off angular frequency at which the resistive part of Z t becomes zero. Above 
this frequency, the tunnel diode cannot amplify. a» x is the self-resonant 
angular frequency where the imaginary part of Z t becomes zero. From 
Eq. (11-13), 



w B 



U) X 



I G G*\K Gl 1 A* , f1 1/1x 

= m t -c>) = cfe- 1 ) (ll - l4) 

= /J__^Y 2 = ^^-lf (11 _i5) 

\L t C C 2 ) C\G 2 L t V K } 



The system stability may be investigated by determining the distribution 
of the poles and zeros in the complex S-plane. Writing Z t in terms of the 
complex frequency, S, we have 

ym _ S 2 L t C + (R t C - L t G)S + 1 - Rfi ml ^ 

Z t {S) - SC-G {U } 



270 



THE TUNNEL DIODE 



CHAP. II 



e-S*B'(S-S)'-T2T <"-" 



The zeros are given by 

s-- 1 - 

6_ 2 

The condition for stability is that the zeros are in the left-half side of the 
S-plane. From Eq. (1 1-17), S will have a negative real part only if 

u 
1 



~>l 



and 



*«< 



(11-18) 
(11-19) 



+ Imaginary 



Z-plane 




FIG. 11-8. Nyquist diagram for the tunnel diode circuit. 



Thus the conditions for stability may be written in the form 

G >R > >J C 



(11-20) 



The stability conditions when substituted in Eqs. (11-14) and (11-15) give 

< oj r < oj x (11-21) 

The Nyquist plot of this case is shown in the lower curve of Fig. 1 1-8. 
Physically, we can say that the negative conductance is an energy source 
which becomes zero at the resistance cut-off angular frequency, cu R . For 
stability, this angular frequency must be real and positive (a> R > 0) and less 
than the angular frequency where the reactance of the circuit becomes zero 

(co R < U) X ). 

Equations (1 1-18) and (1 1-19) give the two requirements for tunnel diode 
amplifier stability. In terms of the total circuit values, R t and L u the expres- 
sions can be rewritten to give 



*<£ 



(11-22) 



SEC. 11-4 



and 



THE TUNNEL DIODE AS A CIRCUIT ELEMENT 



u < 



R t C 



271 

(11-23) 



It is of interest to see what will happen if these requirements are not 
fulfilled. For instance, if R t > l/G, a> R is no longer real, and the device 
cannot be used as an amplifier but may be used as a switch. When L t > 



50 ohm line 



Tunnel 
diode 



16 Q 0.05-0.1 (xh 
A/VV r fftfu" v 



50 ohm line 




Load line 



FIG. 1 1-9. Circuit of 100 Mc/sec amplifier. 

CRJG, oj r > oj x and the circuit becomes a nonlinear relaxation oscillator. 
This case is shown in the upper curve of Fig. 1 1-8. Sinusoidal oscillations 
occur when co R = o» x , i.e., when L t = CRJG. 

Figure 11-9 shows the circuit of an amplifier having a bandwidth of 
20 Mc/sec centered on 100 Mc/sec and a gain of 32 db. This shows the 
use of the tunnel diode as an amplifier at high frequencies. 

In Fig. 1 1-10, a d-c load line has been drawn on the tunnel diode charac- 
teristic. The load line corresponds to 
the case where the total d-c load 
Ri < l/G, and there is only one inter- 
section with the device characteristic. 
For the tunnel diode to operate reliably 
in this condition, the intersection point 
must be kept constant. Thus, E x must 
be accurately controlled. 

The load line drawn on Fig. 11- 
11(a) shows the case when R 2 > l/G. 
This corresponds to the circuit of Fig. 
11-1 1(b). There are three intersection 
points on the tunnel diode characteris- 
tic, A, B, and C. A and C are stable 
and B unstable. When a trigger voltage 
is placed in series with the supply, 
E 2 , point A rises up the diode characteristic until it reaches the peak of the 
curve at a current I p and a voltage V p . The operating point then follows the 




FIG. 11-10. Tunnel diode characteristic 
with load line where the load resistance 
is less than — l/G. 



272 



THE TUNNEL DIODE CHAP. II 




Trigger 
voltage 



R 2 



Tunnel 
diode 



(b) 

FIG. 11-11. Tunnel diode characteristic and circuit when used as 
a switch. 

path shown by the dashed line. Thus the tunnel diode switches from A to C 
A switching time of less than 10 ~ 9 sec can be achieved by a circuit of this type. 



BIBLIOGRAPHY 

Quantum mechanical tunneling 

Bohm, David, Quantum Theory, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 
1951 

Van Name, F. W. Jr., Modern Physics, 2d ed., Englewood Cliffs, N.J.: 
Prentice-Hall, Inc., 1962 

Tunnel diodes 

Davidsohn, U.S., Y. C. Hwang, and G. B. Ober, "Designing with Tunnel 
Diodes," Electronic Design, 8, 3 (February 3, 1960), 50-55; 8, 4 (February 
17, 1960), 66-71 

Hall, R. N., "Tunnel Diodes," I.R.E. Transactions on Electron Devices, 
ED-7, 1 (January 1960), 1-9 

Lesk, I. A. and J. J. Suran, "Tunnel Diode Operation and Application," 
Electrical Engineering, 79, 4 (April 1960) 

Todd, C. D., "Tunnel Diode Applications," Electrical Engineering, 80, 4 
(April 1961), 265-271 



PROBLEMS 

1 1- 1 Derive the value of Tas given in Eq. (11-11). 

1 1-2 Calculate T for U - E = 1 ev where the particles are electrons and 
a = 100 A. 



PROBLEMS 273 

1 1-3 Draw the S-plane, Z-plane diagrams and a rough plot of R and X 
versus frequency for the three cases : cu x < a> R , w x = w R , and co x > co R . 

1 1-4 Analyze the circuit of Fig. 11-9 showing that amplification will be 
obtained at a frequency of 100 Mc/sec and that stability is attained when 
L is less than 0.084 microhenries. For the tunnel diode: R s = 2 ohms, 
C = 5 pf, L s = 5 x 10~ 9 h, -G = -0.007 mhos. 



APPENDIX 



Useful constants 



Electronic charge e 
Electronic mass m 
Planck's constant h 
Boltzmann's constant k 
No. of atoms of germanium per cc 
Permittivity of free space e 
Relative permittivity of germanium 
Relative permittivity of silicon 



1.602 x 10- 19 coulombs 

9.1085 x 10- 31 kg 

6.625 x 10- 34 joule sec 

1.380 x 10- 23 joule K~ 1 

4.42 x 10 22 

(36tt x 10 9 ) - 1 farads m- 1 

16 

12 



Index 



Abrupt junctions, 72 
a-c operation, 139-142 
Acceptor state, 34 
Active circuits, 144 
Alloy-junction transistors, 129-130 
Alpha-cutoff frequency, 194 
Alternating-current gain, 127 
Amplifiers, electronic devices as, 136-139 
Amplifiers, single and multistage, 167-196 

biasing circuits, 178-182 

CE circuit, 167-196 

characteristic curves, 168-170 

class A, B, and C operation, 183 

frequency response of multistage, 194 

limitations of power amplifiers, 187-189 

load line, 170-172 

multistage, 191-193 

operating point temperature stability, 172— 
178 

power amplifiers, 182-186, 187-189 

thermal runaway, 189-191 
Atomic binding, and valence, 17-18 
Atomic number, 15 
Atomic structure, wave-mechanical theory, 

14-15 
Atomic table, 15-17 

Atomic table for 36 elements (table), 16 
Avalanche breakdown, 86 



(3 (beta), 128 

Band theory, energy levels in crystals, 20-23 

Bardeen, 113 

Base charge control, concept of, 249 

Base region, current flow across, 117-123 

Base region, recombination in, 121 

Base spreading resistance, 148, 203-204 

Base width modulation, 120-121, 208-211 

Beta-cutoff frequency, 194, 234 

Bias, 137 

Bias, in junctions, 79 

Biasing, fixed-base, 176 

Biasing circuits, 178-182 

Body-centered cubic cell, 19 

Bohr, Niels, hydrogen atom theory, 5-7, 10, 

11, 14, 15, 31, 32, 34 
Boltzmann factor, 48, 52, 99-100, 223, 264 
Brattain, 113 
"Built-in" field, 220-221, 223 



CE circuit, 126-128, 167-196 
Characteristics, static, 137 
Charge control, 249-254 
Charge depletion region, 73 
Charges, mobile and immobile, 58-61 
Collector-to-base cut-off current, 121-122 
Collector current, 115-116 
Collector multiplication, 187 
Collector multiplication factor, 202-203 
Common base circuit, 123-126 
Common base connection, 236-238 
Common emitter circuit (CE), 126-128, 

167-196 
Common emitter connection, frequency re- 
sponse in, 233-235 
Conduction band, 22 
Conductors, 25 
Conductors, defined, 4 
Contact potential, 72 
Continuity equation, 89-111, 199-200 

abrupt p-n junctions with constant current, 
94-99 

base region for direct currents, 200-202 

base region for small alternating signals, 
204-208 

charge densities at edge of depletion re- 
gions, 99-101 

derivation, 90-94 

diffusion of charge, 91-94 

drift of charge in electric field, 91 

Einstein equation, 108-109 

electrons, equation for, 94 

electrons and positive holes, 91 

forward-biased abrupt p-n junctions, 104- 
107 

incremental resistance of forward-biased 
p-n junction, 108 

p-n junctions, 107 

positive holes, equation for, 93 

reverse-biased abrupt p-n junctions, 101- 
104 
Covalent binding, 18 
Crystal detectors, early, 112-113 
Crystal unit cells, 19-20 
Crystals, 18-20 

Crystals, electronic mobility, 28-29 
Current amplifier, transistor as, 139 
Current gain, 124-128 
Current negative feedback, 178 
Cut-off current, 122 

276 



Davisson, 12 

d-c conditions, 133-196 

De Broglie, 12 

Degeneracy, 264 

Diamonds, 19, 20, 22, 23, 24 

"Diffused-base transistor," term, 223 

Diffusion, 91-94 

Diffusion, current flow by, 117-119 

Diffusion capacitance, 122-123, 213 

Diffusion length, for electrons, 99 

Diffusion length, for positive holes, 98 

Diffusion techniques, in manufacturing, 130- 

131 
Diffusion transistor, term, 220 
Dirac, 12, 14 
Distribution of power dissipation (table). 

184 
Donor state, 32-33 
Doping, 35 

Drift of charges, in electric fields, 91 
Drift transistor, 220-223 



Effective mass, concept of, 47 

Effective width, of junctions, 73 

Einstein equation, 108-109, 219, 222 

Ekiss, J. A., 254 

Electron energy levels, in crystals, 20-23 

Electron volt, 10 

Electronic mobility, in crystals, 28-29 

Electrons: 

continuity equation for, 94 

diffusion length for, 99 

energy states, 15 

and positive holes, 26-28, 91 
Elements, atomic table, 15-17 
Emitter current, 115 
Emitter efficiency, 119, 202 
Emitter follower, 162 
Emitter resistance, 213 
Energy bands, density of states, 44-47 
Energy density of states, defined, 47 
Energy levels in crystals (band theory), 20- 

23 
Energy state densities, and temperature, 47- 

51 
Energy states, 15, 44-47 
Epitaxial films, 226-227 



Epitaxial mesa transistor, 227 
Esaki, 260 

Exclusion principle, Pauli, 17, 45, 48 
Extrinsic semiconductors, 26 



Fermi level, 48-58, 68-70, 72, 76, 100, 264- 

267 
Fermi-Dirac statistics, 43, 70, 264 
Fixed-base biasing, 176 
Flicker noise, 223-224 
Forbidden gap, 22, 25, 26, 27, 31, 44, 52, 53, 

54, 56, 58, 128 
Forward bias, 79 
Frequency response, 229-259 

charge control, 249-254 

common base connection (simple model), 
236-238 

common emitter connection, 233-235 

components of charge, 251-253 

hybrid-jt circuits, 230-231, 233-236, 238- 
242, 244-247 

iterative common emitter stage, 235-236 

Miller effect, 231-232 

saturation, 242-244, 247-249 

switching, 244-247 

switching times, defined, 247 

switching times for saturation, 254-258 



Gain-bandwidth product, 234 

Gallium arsenide, 267 

Generation, of electron-hole pairs, 35^40 

Generation rate, 39 

Germanium, 4, 18, 25, 31, 33, 128 (also 
passim ) 

Germanium and silicon, common donor and 
acceptor elements (table), 32 

Germanium and silicon, properties of (ta- 
ble), 28 

Germer, 12 

Goudsmit, 14 

Graded junctions, 72 

Group 4 materials, 30 

Group 5 substances, 30 

Grown-junction transistors, 129 

277 



H 



h, Planck's constant, 6, 14 
h parameter equivalents of CE and CC cir- 
cuits (table), 160 
h parameter representation, low frequency, 
154-166 

analysis of general equivalent circuit, 156— 
159 

common base circuit, 161-162 

common collector circuit, 162-163 

common emitter circuit, 163 

current gain, 158 

input resistance, 157 

output resistance, 158 

power gain, 158 

relationship of three sets of, 159-160 

small-signal, 155-156 

three configurations, comparison of, 161— 
163 

variation with I E and V CE , 163-165 

voltage gain, 157 
h parameters, equivalent circuit using, 149- 

150 
Hall effect, 61-64 
Heat-sink, 191 
Heisenberg, 12, 13-14,45 
Hybrid (h) parameters, defined, 149 (see 

also h parameter representation) 
Hybrids circuit, 135, 143, 148-149, 200, 

211-217,229-259 
Hydrogen atom: 

Bohr's theory of, 5-7 

energy of electron orbits, 7-9 

energy level representation, 10-11 

simple theory of, 4-5 

transitions between orbits, 9-10 



Impedance terms, 141 

Impurities, 18, 25-26, 29-35, 58 

Incremental resistance, 108 

Insulators, 24 

Insulators, defined, 4 

Intrinsic germanium, 60-61, 77 

Intrinsic material, 29 

Intrinsic semiconductors, 25, 34, 51-53 

Intrinsic transistor, 203-204 

Ionic binding, 18 

Ionization energy, 10 

Iterative common emitter stage, 235-236 



Junction transistors, 112-133 
Junctions: 

between metals, 70-72 

bias, 79 

breakdown region of p-n junctions, 86 

capacitance, 79-82 

effective width, 73 

flow of current across a p-n junction, 82- 
86 

incremental capacitance, 81 

nonlinear incremental capacitance, 81 

rectifier equation, 82-86 

with applied voltage, 79-82 

with no applied voltage, 72-78 



Lattice constant, 20 

Lifetimes, 38 

Load line, 170-172 

Low frequencies, 133-196 

Low frequency measurements, 150-153 



M 



Majority carriers, 33, 34 
Manufacture: 

alloy-junction transistors, 129-130 

diffusion techniques, 130-131 

grown-junction transistors, 129 

materials, 128-129 

point-contact transistors, 129 

zone refining, 128 
Mass action, law of, 58-59 
Matter, wave nature, 11-14 
Maxwell-Boltzmann statistics, 52 
Mesa diffused-base transistors, 130-131, 227 
Metals, junctions between, 70-72 
Metals, work function, 68-70 
Miller effect, 231-232 
Minority carrier lifetimes, 38 
Minority carriers, 33, 34 
Mobility of electrons and holes, 28-29, 36, 

37 
Modulation noise, 223 
Momentum space diagram, 44, 46 
Multistage amplifiers, 191-193 (see also 
Amplifiers) 

278 






npn transistors, 113-116 

n- and p-type semiconductors, 29-35 

n-type, term, 33 

«-type germanium, 104 

«-type material, 97-98 

>?-type semiconductors, 54-55 

Noise, 223-225 

Noise figure, defined, 224 

Norton, 125, 146, 149, 150 

Numbered and lettered h parameters (table), 

155 
Nyquist plot, for tunnel diode, 270 



Ohmic contacts, 107 

Operating point temperature stability, 172- 
178 



it circuit, 148-149 

Parameters, z, y, and h, 135, 141, 142-144 

"Passivate," term, 226 

Passive circuits, 144 

Pauli exclusion principle, 17, 45, 48 

Perfect crystals, 18 

Performance quantities for CB, CE, and CC 

circuits (table), 147 
Performance quantities for h parameter 

equivalent circuit (table), 158 
Photon, 10 
Physical characteristics, 199-228 

alpha, components of, 202-203 

base width modulation, 208-211 

continuity equation {see Continuity equa- 
tion) 

drift transistor, 220-223 

epitaxial films, 226-227 

hole density in base region, 200-202 

hybrid-jt; representation, 211-217 

intrinsic transistor, 203-204 

noise, 223-225 

punch-through, 226 

small alternating signals, 204-208 

surface effects, 225-226 

transmission line analogy, 208-211 

variation of alpha with emitter current, 
217-220 
Planck, Max, quantum theory, 5-7, 14 



p- and n-type semiconductors, 29-35 
p-n junctions: 

breakdown region, 86 

conditions of operation, 96 

continuity equation {see Continuity equa- 
tion) 

discussion of, 107 

flow of current across, 82-86 

as rectifiers, 35 
pnp transistors, 116-117 

common base circuit, 123-126 

common emitter circuit, 126-128 

current flow across base region, 117-123 

hole density in base region, 200-202 
Point-contact germanium diode, development 

of, 113 
Point-contact transistors, development of, 

113 
Point-contact transistors, manufacture of, 

129 
Poisson equation, 74-76 
Positive holes: 

continuity equation for, 93 

diffusion length for, 98 

effective mass of, 47 

and electrons, 26-28 
Potential barriers, 78 
Power amplifiers, 182-186 
Properties of tunnel diodes (table), 267 
p-type germanium, 104 
p-type semiconductors, 55-56 
Punch-through, 187-188, 226 
Punch-through voltage, 226 



Quantum mechanical tunneling, 261-263 
Quantum numbers, 14-15 
Quantum theory, 5-7 



Radiation, relation to matter, 11-14 

Recombination, in base region, 121 

Recombination, of electron-hole pairs, 35-40 

Rectifier equation, 82-86 

Relationship between parameters (table), 152 

Resistance noise, 223 

Reverse bias, 79 

Rise time, defined, 247 

Rutherford, 4 



279 



Saturation, 188, 242-244, 247-249 

current density, 83 

switching times for, 254-258 
Schrodinger, 12-13, 15, 261, 263 
Semiconductors: 

defined, 4 

electrical conduction in, 3-42 

electrons and holes, 43-66 
Shockley, 113 
Shot noise, 223 

Silicon, 4, 18, 25, 128-129 (also passim) 
Simmons, C. D., 254 
Single crystals, 18 
Solid-state devices, 4 
Sommerfeld, 11, 68 
Space charge region, 73 
Speed-up capacitors, 257-258 
"Spot" noise figure, 225 
Stability factor equation, 177 
Step junctions, 72 
Storage time, 244 
Surface effects, 225-226 
Surface leakage, 187 
Surface recombination velocity, 225-226 
Switching, 244-247 
Switching times, defined, 247 
Switching times, for saturation, 254-258 



Tunnel diode (con't) 
limitations, 268 
Nyquist plot, 270 
oscillation frequencies, 268-269 
quantum mechanical tunneling, 261-263 
resistance cut-off angular frequency, 269 
self-resonant angular frequency, 269 
switching times, 268 
Two port network, transistor as, 135-153 
Typical h parameter values (table), 161 



U 



Uhlenbeck, 14 

Uncertainty principle (Heisenberg), 13-14, 

45 
Unfilled energy states, 33 
Unit momentum cells, 45 



Vacuum level, 69 

Valence, and atomic binding, 17-18 

Valence, Fermi level, and work function for 

five metals (table), 70 
Valence band, 22 

Variation in I CB0 , h FE , and I B (table), 175 
Voltage gain, 125, 126 



T circuit, 144-147 
Temperature: 

and energy state densities, 47-51 

limit placed by, 56-57 

stability, 172-178 

variation of Fermi level, 56-57 
Thermal agitation, 223 
Thermal equilibrium values, 35 
Thermal resistance, 190 
Thermal runaway, 172, 189-191 
Thermionic emission, 70 
Thevenin circuits, 146, 149, 150 
Thomson, G. P., 12 
Transmission line analogy, 208-211 
Transport factor, 202 
Traps (recombination centers), 38 
Tunnel diode, 260-273 

as circuit element, 268-272 

energy bands, 263-265 

forward and reverse characteristics, 265- 
268 



W 



Wave packet, 13 

Wave (quantum) mechanics, 11-14 

"White" noise, 223 

Work function, of materials, 69 

"Worst-case" design, 238 



y parameters, equivalent circuit using, 148- 
149 



z, y, and h parameters, 135, 141, 142-144 
z parameters, equivalent circuit using, 144- 

147 
Zener breakdown, 86 
"Zener" devices, 86 
Zone refining, 128 

280 



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